Everything you need to ace pre-algebra and algebra 1 in one big fat notebook

Chapter 67 GRAPHING QUADRATIC FUNCTIONS A QUADRATIC FUNCTION is a function that can be writ ten in the standard form: y = ax2 + bx + c, where a = 0 Every quadratic function has a U-shaped graph called a parabola. There are several characteristics of the graph that we can calculate from the equation. 588

DIRECTION Most parabolas open up either upward or downward. The parabola opens upward if the value of a is positive. The parabola opens downward if the value of a is negative. If a > 0, then the parabola opens UPWARD. This graph of y = x2 opens upward because a = 1. If a < 0, then the parabola opens DOWNWARD. This graph of yop=en-4sxd2o-w2n0wxa-rd947 because a = -4. To determine the direction of the parabola y = 5x2 - 2x + 1: Write the formula y = ax2 + bx + c. Write the values for each variable: a = 5, b = -2, and c = 1 a > 0, so the parabola opens upward. 589

EXAMPLE: Determine the direction of the 3 parabola y = - 2 (x - 8)2. Rewrite the equation into the form y = ax2 + bx + c, y = - 3 (x - 8)2 2 = - 3 (x 2 - 16x + 64) 2 = - 3 x2 + 24x - 96 2 Since ado=w-n2w3 a, trhdi.s means that a < 0, so the parabola opens VERTEX The VERTEX of the parabola is the “tip” of the parabola. You can find the coordinates of vertex the vertex of the graph of y = ax2 + bx + c. The x-coordinate of the vertex is at: - b . 2a The y-coordinate of the vertex is found by substituting the value of the x-coordinate back into the equation. 590

EXAMPLE: Find the coordinates of the vertex of the graph of y = x2 + 6x - 7. Step 1: Find the x-coordinate by using the formula x = - b . 2a Since a = 1, b = 6, and c = -7, the x-coordinate of the vertex b 6 is: x=- 2a = - 2• 1 = -3. Step 2: Find the Notice that this parabola y-coordinate by opens upward substituting the value because of the x-coordinate back a > 0. into the equation. Since the x-coordinate is x = -3, the y-coordinate is: y = (-3)2 + 6(-3) - 7 = -16. Therefore, the coordinates of the vertex are: (-3, -16). vertex 591

( )EXAMPLE: Find the coordinates of the vertex of the32 + 5. graph of y = -4 x - 2 Rewrite the equation into the form y = ax2 + bx + c. ( )y = -4 x - 3 2 2 +5 ( )= -4 9 x2 - 3x + 4 +5 = -4x2 + 12x - 9 + 5 = -4x2 + 12x - 4 Step 1: Find the x-coordinate by using the formula x = - b . 2a Since a = -4, b = 12, and c = -4, the x-coordinate of the b 12 3 vertex is: x = - 2a = - 2 • (-4) = 2 Step 2: Find the y-coordinate by substituting the value of the x-coordinate back into the equation. Since the x-coordinate is x = 3 , the y-coordinate is: 2 (( ) )y = -4 3 - 3 2 2 2 +5=5 592

Therefore, the coordinates of the vertex are: ( )3, 5 . 2 Notice that this parabola opens downward because a < 0. VERTEX FORM The standard form of a quadratic equation is: y = ax2 + bx + c. Another form of writing a quadratic equation is the vertex form. In both the standard form and vertex form y is the y-coordinate, x is the x-coordinate and a is the constant that tells when the parabola is facing up (> 0) or down (< 0). Vertex Form y = a(x - h)2 + k, where (h, k) are the coordinates of the vertex. EXAMPLE: Find the coordinates of the vertex of the quadratic equation y = 7(x - 3)2 + 8. Since a = 7, h = 3, and k = 8, the coordinates of the vertex are: (h, k) = (3, 8). 593

EXAMPLE: Find the coordinates of the vertex of the 1 3 quadratic equation y = - 2 (x + 7)2 + 4 . Don’ t forget to include the negative sign. Since a = - 1 , h = -7, and k = 3 , the coordinates of the 2 4 vertex are: ( )(h, k) =-7,3 4 To rewrite a quadratic equation from standard form to vertex form, use the Completing the Square method, because the vertex form contains a perfect square. EXAMPLE: Rewrite the equation y = x2 + 10x - 7 into vertex form. y = x2 + 10x -7 Rewrite the equation so the RHS has the form x2 + bx. y + 7 = x2 + 10x 594

This means that b = 10: Step 1: Calculate the value of b : 10 =5 2 2 Step 2: Square that value: 52 = 25 Step 3: Add that number to both sides of the equation: y + 7 + 25 = x2 + 10x + 25 y + 32 = x2 + 10x + 25 Step 4: Factor the RHS and then solve for y: y + 32 = (x + 5)2 y = (x + 5)2 - 32 EXAMPLE: Rewrite the equation y = 3x2 - 6x + 7 into vertex form and name the coordinates of the vertex. y = 3x2 - 6x + 7 y - 7 = 3x2 - 6x 1 y - 7 = x2 - 2x 3 3 595

This means that b = 2: Step 1: Calculate the value of b : -2 = -1 2 2 Step 2: Square that value: (-1)2 = 1 Step 3: Add that number to both sides of the equation: 1 y - 7 + 1 = x2 - 2x + 1 3 3 1 y - 4 = (x - 1)2 3 3 Step 4: Factor the RHS and then solve for y: 1 y = (x - 1)2 + 4 3 3 y = 3(x - 1)2 + 4 Therefore, the coordinates of the vertex are: (1, 4). 596

MAXIMUM/MINIMUM The minimum value of an If a parabola opens upward, upward-opening it has a MINIMUM VALUE, parabola is at and that minimum value is the vertex. at the vertex. If a parabola opens The maximum downward, it has a value of a MAXIMUM VALUE, and downward- that maximum value is opening parabola at the vertex. is at the vertex. The maximum/minimum value is always the y-coordinate of the vertex. To find the maximum/minimum value of a quadratic equation: Step 1: Determine if the parabola opens upward or downward. upward = minimum value downward = maximum value Step 2: Find the coordinates of the vertex. The y-coordinate of the vertex is the minimum/maximum value. 597

EXAMPLE: Find whether y = x2 - 8x + 3 has a maximum value or a minimum value. Then find that value. Step 1: Since a = 1, this means that a > 0, so the parabola opens upward. The parabola has a MINIMUM value at the vertex. Step 2: Since a = 1, b = -8, and c = 3, the x-coordinate of the vertex is: b -8 x=- 2a = - 2•1 = 4. Substitute x = 4 into the Minimum Value of -13 equation. The y-coordinate of the vertex is: y = (4)2 - 8(4) + 3 = 16 - 32 + 3 = -13. The parabola y = x2 - 8x + 3 has a MINIMUM VALUE of -13. 598

AXIS OF SYMMETRY All parabolas are SYMMETRICAL. axis of symmetry This means that if we draw a vertical line down the center of the parabola, the left side and right side would be mirror images of each other. That vertical line is called the AXIS OF SYMMETRY. The equation of the axis of symmetry is x = - b . 2a This is the same value as the x-coordinate of the vertex, because the axis of symmetry goes through the vertex. EXAMPLE: Find the equation of the axis of symmetry of 1 the parabola y = - 2 (x - 1)(x + 8). Rewrite the equation into the form y = ax2 + bx + c. y = - 1 (x - 1)(x + 8) 2 = - 1 (x 2 + 7x - 8) 2 =- 1 x2 - 7 x + 4 2 2 599

Since a = - 1 ,ebqu=a-tio27n , and c = 4, t h2e of the axis of symmetry is: x=- b = - (- 7 ) ) 2a 2 2 • (- 1 2 = - 7 2 Axis of symmetry: x = - 7 2 INTERCEPTS INTERCEPTS are the points where the parabola intersects- or intercepts- the x-axis and the y-axis. They are expressed as numbers or coordinates. The x-intercept is where the parabola intersects the x-axis. The y-intercept is where the parabola intersects the y-axis. To find any y-intercept, substitute 0 for x and then solve for y. To find any x-intercept, substitute 0 for y and then solve for x. 600

An x-intercept is also known as a ZERO or a ROOT. EXAMPLE: Find all the intercepts of the parabola y = x2 + 3x - 10. Find the y-intercept by substituting 0 for x: y = x2 + 3x -10 = (0)2 + 3(0) - 10 = -10 Therefore, the y-intercept is -10 or (0, -10). Find the x-intercepts by substituting 0 for y: y = x2 + 3x - 10 (0) = x2 + 3x - 10 Use factoring or Completing the Square or the Quadratic Formula to find the solutions. 0 = (x + 5)(x - 2) x = -5, 2 601

Therefore, the x-intercepts x-intercepts are: -5 and 2 or (-5, 0) and (2, 0). y-intercept THE DISCRIMINANT Some parabolas will have 2 x-intercepts: Some parabolas will have only 1 x-intercept: Some parabolas will have 0 x-intercepts: 602

We can find the amount of x-intercepts (or roots) by analyzing the value of the DISCRIMINANT: D = b2 - 4ac If D > 0, the parabola has 2 x-intercepts. If D = 0, the parabola has 1 x-intercept. If D < 0, the parabola has 0 x-intercepts. EXAMPLE: For the parabola y = 3x2 - 6x + 5, find the value of the discriminant. Then determine the amount of x-intercepts that the parabola has. Since a = 3, b = -6, and c = 5, the value of the discriminant is: D = b2 - 4ac = (-6)2 - 4 • 3 • 5 = 36 - 60 = -24 Since D < 0, the parabola has 0 x-intercepts. 603

To graph a quadratic equation, take all the different characteristics of a parabola and connect the points. When graphing a parabola, we should always find: The direction of the parabola. The coordinates of the vertex. The coordinates of the intercepts-y-intercept and x- int ercept (s). The discriminant will tell how many x-intercepts there are. EXAMPLE: Graph y = x2 - 4x - 5. Find the direction of the parabola: a = 1, and since a > 0, the parabola opens upward. Find the coordinates of the vertex: Since a = 1, b = -4, and c = -5, the x-coordinate of the vertex is: 604

x=- b = - -4 =2 2a 2•1 Substitute x = 2 into the equation. The y-coordinate of the vertex is: y = (2)2 - 4(2) - 5 = 4 - 8 - 5 = -9 Therefore, the coordinates of the vertex are: (2, -9) Find the y-intercept by substituting 0 for x: y = x2 - 4x - 5 = (0)2 - 4(0) - 5 = -5 Therefore, the y-intercept is -5 or (0, -5). Find the number of x-intercepts by calculating the discriminant. Since a = 1, b = -4, and c = -5, the value of the discriminant is: D = b2 - 4ac. = (-4)2 - 4 • 1 • (-5) = 16 + 20 = 36 Since D > 0, the parabola has 2 x-intercepts. 605

Find the x-intercepts by substituting 0 for y: y = x2 - 4x - 5 (0) = x2 - 4x - 5 Use factoring to find the solutions. 0 = (x - 5)(x + 1) x = 5, -1 Therefore, the x-intercepts (-1, 0) (5, 0) (0, -5) are: 5 and -1 or (5, 0) and (-1, 0). Graph the points and connect them by drawing a parabola. (2, -9) 606

w For problems 1 through 3, find the following characteristics of each parabola: • the direction • the coordinates of the vertex • whether the parabola has a maximum or minimum value • the equation of the axis of symmetry • the value of the discriminant • the coordinates of all intercepts 1. y = x2 - 6x + 8 2. y = -x2 + 2x + 3 3. y = 2x2 + 8x - 10 607

For problem 4, rewrite the quadratic equation into vertex form. Then state the coordinates of the vertex. 4. y = 4x2 + 12x - 7 5. Draw the graph of y = x2 - 2x - 8. 6. Draw the graph of y = -2x2 - 8x - 3. 608

1. direction: upward vertex: (3, -1) the parabola has a minimum value axis of symmetry: x = 3 discriminant: D = 4 y-intercept: (0, 8); x-intercepts: (4, 0) and (2, 0) 2. direction: downward vertex: (1, 6) the parabola has a maximum value axis of symmetry: x = 1 discriminant: D = 16 y-intercept: (0, 3); x-intercepts: (3, 0) and (-1, 0) 3. direction: upward vertex: (-2, -18) the parabola has a minimum value axis of symmetry: x = -2 discriminant: D = 144 y-intercept: (0, -10); x-intercepts: (-5, 0), and (1, 0) ( )4. 3 2 16; 3 y=4 x+ 2 - vertex is ( 2 , -16) More answers 609

5. 6. 610

Chapter 68 SOLVING QUADRATIC EQUATIONS BY GRAPHING When we graph the quadratic equation y = ax2 + bx + c, the x-intercepts are where the parabola crosses the x-axis and the intercepts have a y-value of 0. This is why the x-intercepts represent the roots or the solution of the quadratic equation. 0 = ax2 + bx + c or ax2 + bx + c = 0 Notice that y has been replaced with 0. 611

EXAMPLE: Find the solutions of - 2 x2 - 2 x + 4 = 0. 3 3 Use the graph of y = - 2 x2 - 2 x + 4, shown. 3 3 The tshoeluxti-oinnsteorfce-pt23s x2 - 2 x+4=0 are 2 2 of t h3e equation 3 3 y=- x2 - x + 4. Since the x-intercepts of the graph are -3 and 2, the solutions are: x = -3 and x = 2. EXAMPLE: Find the solutions of x2 - 3x + 4 = 0. Use the graph of y = x2 - 3x + 4. The solutions of x2 - 3x + 4 = 0 are the x-intercepts of the equation y = x2 - 3x + 4. Since there are no x-intercepts, there is no solution. 612

EXAMPLE: The vertex of a parabola is at (-4, -3). One of the roots of the quadratic equation is (-6, 0). Find the other root of the quadratic equation. Graph the vertex and the root, where the axis of symmetry runs through the vertex. Using symmetry, we are able to find the other root, because the other root is an equal distance away from the axis of symmetry. Therefore, the other root is: (-2, 0). 613

There are real-life applications to finding solutions to quadratic equations. EXAMPLE: Jamie kicks a soccer ball into the air, away from her. The path that the ball takes is in the shape of a parabola and is represented by the equation y = 2x2 + 9x, where x represents how far away the soccer ball travels (in meters), and y represents how high the soccer ball travels above the ground (in meters). How far away is the soccer ball when it hits the ground? Since we are examining the moment when the soccer ball hits the ground, this means that the height of the soccer ball above the ground is y = 0. This is like finding the x-intercept of the graph of a quadratic equation. 614

Set ting y = 0 for y = 2x2 + 9x: (0) = 2x2 + 9x Use factoring. 0 = -x (2x - 9) Apply the Zero-Product Principle. x = 0 or 2x - 9 = 0 x=0 or x= 9 2 There are two answers: 0 meters away, or 9 = 4.5 meters 2 away. The first answer doesn’t make sense (it represents where the ball is before it is kicked), so the answer is: 9 2 = 4.5 meters away. 615

w 1. Find the solution(s) of -x2 + 6x - 5 = 0, using this graph of y = -x2 + 6x - 5 . 2. Find the solution(s) of 2 8 8 3 x2 + 3 x + 3 = 0, using this graph of 2 8 8 y= 3 x2 + 3 x + 3 . 3. Find the solution(s) of -x2 - 4 = 0, using this graph of y = -x2 - 4. 616

4. Find the solution(s) of 1 x2 + 1 x - 3 = 0, using this graph 1 1 8 4 of y = 8 x2 + 4 x - 3. 5. The vertex of a parabola is at (2, -5). One of the roots of the quadratic equation is (8, 0). Find the other root of the quadratic equation. 6. A cannonball is fired. The path the ball takes is in the shape of a parabola and is represented by the equation y = -4x2 + 31x, where x represents how far away the cannonball travels (in miles) and y represents how high the cannonball travels above the ground (in miles). How far away is the cannonball when it hits the ground? answers 617

1. x = 1 or x = 5 2. x = -2 3. No solution 4. x = -6 or x = 4 5. (-6, 10) 6. 7.75 miles away 618

INDEX A order of operations, central tendency, 17-22 measures of, 335-338 absolute value, 17, 27-28, 52-53, 61-63 types of numbers, change, percent rate of, 2-6 131-134 actions, 364 addition Associative Property coefficient, 156-157 of Addition, 10 collecting like terms. See Associative Property of Multiplication, 10-11 of, 10 combining like terms axes combinations, 385, 390-391 Commutative Property description of, 248 combining like terms, of, 9, 11 x-axis, 248, 277-278, 600 169-174, 190, 429 of decimals, 59-61 y-axis, 248, 277-278, commissions, calculating, Distributive Property 600 118-119 of Multiplication over, axis of symmetry, Commutative Property 12-13, 14 599-600 order of operations of Addition, 9, 11 and, 17-20 B of Multiplication, 9-10, of polynomials, balance, 125 428-436 base, exponents and, 171 of positive and complement of an event, negative fractions, 142-146 51-52, 54-56 binomial, 156, 428-429, 456 371-372 of positive and box plots, 348-352 completing the square, negative whole braces, for sets, 252 numbers, 24-29 561-570, 594 of radicals, 521-526 C compound events, 375-384 addition method, 241 calculating discounts, compound inequalities, additive inverse, 33-34, 53 algebraic expressions, 111-115 206-215 evaluating, 163-165 calculating markups, compound interest, 125 algebraic properties, 9-14 constant, 158 alphabetical order, 159-160 116-118 constant of arithmetic properties calculating percent, algebraic properties, proportionality, 93 9-16 101-103 constant of variation, 93 calculating sales tax, coordinate plane, 248-252 correlations, 356-357 107-109 counting numbers, 2 counting principle, 369-370 cross multiplication, 90, 92 619

cross products, 90, 92 rationalizing, of polynomials, cross-reducing/cross- 529-530 453-460 canceling, 44 simplifying, 165 of positive and cube roots subtracting fractions negative fractions, 43-50 description of, 510-511 with like, 53 perfect cubes, 511-512 dependent events, 375, of positive and simplifying, 517-518 negative whole cubes 379-381 numbers, 37-42 difference of two cubes dependent variables, of radicals, 527-530 formula, 499-502 177-179, 396-397 divisor, 70 sum of two cubes descending order, 158-160, domain, 252, 396-403 drawings, scale, 79 formula, 499-502 171-172 difference of two cubes e d formula, 499-502 elimination, solving data difference of two squares systems of linear collecting, 329 equations by, definition of, 326 formula, 495-496 237-244 displaying, 343-357 direction of parabola, qualitative, 326, 328-329 elimination method, quantitative, 326-327, 589-590 237-244 329 discounts, calculating, sampling and, 330-332 equations 111-115 definition of, 176 decimals discriminant, 579-586, first-degree, 180-182 adding and subtracting, introduction to, 59-63 602-606 176-184 converting percent to, Distributive Property with one variable, 100 185-194 multiplying and of Multiplication over quadratic, 415-418 dividing, 67-74 Addition, 12-13, 14, 172 solving by elimination, repeating, 6 237-244 terminating, 6 of Multiplication over solving by graphing, Subtraction, 13-14, 299-306 degree, 157 172, 189-190 solving by substitution, denominators 225-234 multiplying and dividing writing in slope- adding fractions polynomials and, intercept form, with like, 51-52 453-460 279-283 adding fractions dividend, 70 with unlike, 54-56 division of decimals, 70-72 of exponents, 437-442 of monomials, 445-450 order of operations and, 17-22 620

See also linear introduction to, 155-160 standard form of equations; lines; standard form of, expression, 158-160 quadratic equations 158-160 standard notation, equivalent fractions, types of, 156 149-152 54-56, 89 f vertex form, 593-596 equivalent linear equations, formulas 302 factorial functions, 387-388 difference of two equivalent ratios, 78 cubes, 499-502 evaluating algebraic factoring solving quadratic difference of two expressions, 163-165 equations by, 541-552 squares, 495-496 events using GCF, 462-474 using grouping, 475-478 factoring using special, complement of, 371-372 using special formulas, 495-502 compound, 375-381 495-504 definition of, 364 when a=1, 481-490 for finding exponents when a≠1, 491-494 discriminant, 579-586 definition of, 142 fractions with negative, factorization, 465, 472 perfect square factors, definition of, 462 trinomial, 497-498 146 first-degree equations, introduction to, 142-148 for permutations, multiplying and 180-182 388-389 FOIL method, 456, dividing, 437-444 for point-slope form, negative, 144-146 482-483, 548 290 power of a power and, form/notation Quadratic Formula, 448-450 interval notation, 573-576 properties of, 143-144 212-215 scientific notation and, rewriting, 219-224 point-slope form, for simple interest, 124 149-152 289-298 for slope, 263, 268, 270, simplifying and, of quadratic equations, 280 437-444 534-535 sum of two cubes, in variable(s), 157-158 expressions scientific notation, 499-502 combining like terms in, 149-152 fractions 169-172 slope-intercept form, adding and subtracting, definition of, 155 277-283, 295, 315-316, 51-58 evaluating algebraic, 319 equivalent, 54-55, 89 163-168 standard form of multiplying and exponents and, 157-158 equation, 294 dividing, 43-50 with negative exponents, 146 percent and, 99 621

functions greatest common factor rate of, 124 application of, 415-421 (GCF), 466-469 simple, 123-130 definition of, 396-397 intersection (and), evaluating, 407-411 grouping, factoring factorial, 387-388 polynomials using, 206-209, 549 nonlinear, 415-426 475-478 intersection point, notation for, 407-414 quadratic, 588-610 h 299-300 relations and, high variability, 329 interval notation, 396-406 histograms, 347-348 212-215 fundamental counting i inverse, additive, 33-36 principle, 369-370 independent events, inverse operations, g 375-378 186-194, 219-224 independent variables, irrational numbers, 4, 5, graphing inequalities, 196-197 177-179, 396-397 508-509 line from table of indexes, 521-524, 527-529 isolating the variable, values, 255-262 inequalities linear inequalities, 185-194 307-314 graphing, 196-197 quadratic functions, graphing linear, l 588-606 solving quadratic 307-314 least common multiples equations by, solving, 198-201 (LCM), 54-56, 221, 611-616 solving compound, 241-243 solving systems of linear equations by, 206-215 like terms, combining, 299-306 solving one-variable, 169-174, 190, 429 solving systems of linear inequalities by, 195-205 line of best fit, 354-355 315-324 solving systems of line plots, 345-346 linear equations graphs linear, by graphing, displaying data using, 315-324 description of, 180-182 343-362 writing, 195 equivalent, 302 of functions, 415-426 inequality symbols, 196, forms of, 295 200, 212 with no solution, 301 gratuities, calculating, infinity symbol, 215 solving systems of by 118-119 integers, 3, 5 intercepts, 277-278, elimination, 237-244 600-602 solving systems of by interest compound, 125 graphing, 299-306 solving systems of by substitution, 225-236 See also lines; slope; slope-intercept form 622

linear functions, 415 multiplication multiplying and linear inequalities Associative Property dividing, 37-42 of, 10-11 graphing, 307-312 Commutative Property subtracting, 33-36 solving systems of, by of, 9-10, 171 negative slope, 267, 272 cross, 90 no correlation, 357 graphing, 315-324 of decimals, 67-69 nonlinear functions, lines distributive properties of, 12-14 415-421 coordinate plane and, of exponents, 437-442 notation/form 248-252 fundamental counting principle and, interval notation, graphing from a table 369-370 212-215 of values, 255-257 of monomials, 445-452 order of operations point-slope form, point-slope form for, and, 17-20 289-295 289-295 of polynomials, 453-460 of positive and of quadratic equations, slope of, 263-276 negative fractions, 534-535 slope-intercept form 43-50 of positive and scientific notation, for, 277-283, 295, negative whole 149-152 315-316, 319 numbers, 37-42 standard form for, 294 of radicals, 527-532 slope-intercept form, low variability, 329 277-288, 295, 315-316, n 319 m natural numbers, 2, 5 standard form of markups, calculating, negative correlation, 356 equation, 294 116-118 negative exponents, standard form of maximum value, 337, 144-145 expression, 158-160 597-598 negative fractions standard notation, mean, 336 adding and subtracting, 149-154 measures of central 51-56 vertex form, 593-596 tendency, 335-338 multiplying and number line method, measures of variation, dividing, 43-50 24-27 339-340 negative numbers number system median, 337-338 adding, 24-32 minimum value, 337, adding and subtracting decimals, 59-63 597-598 mode, 338 adding and subtracting monomials positive and negative fractions, 51-58 definition of, 156, 428 multiplying and adding positive and negative whole dividing, 445-450 numbers, 24-32 623

multiplying and dividing maximum value of, pie charts, 343 decimals, 67-74 597-598 plots, scatter, 353-357 point, intersection, multiplying and dividing minimum value of, positive and negative 597-598 299-300 fractions, 43-47 points, 248-252 as result of graphed point-slope form, 289-298 multiplying and dividing quadratic equation, polynomials positive and negative 415-418, 537, 588-606 whole numbers, adding and subtracting, 37-42 vertex of, 590-598 428-434 PEMDAS (Parentheses, subtracting positive definition of, 156 and negative whole Exponents, degree of, 158 numbers, 33-36 Multiplication, factoring using GCF, Division, Addition, numbers, types of, 2-6 Subtraction), 18 462-474 numerators, simplifying, percent factoring using calculating, 101-103 165 converting to decimals, grouping, 475-478 100 factoring using special o converting to fractions, one-variable inequalities, 99 formulas, 495-504 definition of, 98 multiplying and 195-201 finding whole from, order of operations, 103-104 dividing, 453-460 overview of, 98-106 population, 331 17-22 probability and, positive correlation, 356 ordered pair, 248-249, 363-372 positive fractions percent applications, 353-355 107-119 adding and subtracting, origin, 248-249, 309-310 percent rate of change, 51-58 outcomes, 364 131-134 outliers, 340, 355 perfect cubes, 499, 511-512 multiplying and perfect square trinomial dividing, 43-50 p formula, 497-498 parabolas perfect squares, 508-509, positive numbers 553-557 adding, 24-29 axis of symmetry for, permutation formulas, multiplying and 599-600 388-389 dividing, 37-42 permutations, 385-389 subtracting, 33-36 direction of parabola, 589-590 positive slope, 267, 271 power of a power, 448-450 discriminant and, price 602-606 finding original, 110, 114, intercepts and, 117-118 600-602 sales tax and, 107-109 624

principal, 124 testing solutions for, ratios principal square root, 508 536 definition of, 76 probability equivalent, 79 Quadratic Formula, solving overview of, 76-82 compound events and, quadratic equations probability and, 375-381 with, 573-576 364-368 simplifying, 78 overview of, 363-374 quadratic functions, tables and, 135-140 permutations and graphing, 588-606 real numbers, 4, 5 combinations and, qualitative data, 326, reciprocals, 46-47, 144-146 385-392 328-329 relations proportion, 89-97 proportionality, constant quantitative data, definition of, 252, 396 of, 93 326-327, 329 functions and, 396-406 repeating decimals, 6 q quartiles, 349-352 repetition quotient, 70 permutations with, quadrants, 251 quadratic equations r 385-387 permutations without, discriminant and, radical sign, 506 579-583 radicals 387-389 rise, 263-265, 268-270, 280 form of, 534-535 adding and subtracting, root (x-intercept), 601-602 introduction to, 534-540 521-526 run, 263-265, 280 parabolas as result of components of, 521 s graphing, 415-418, cube roots, 510-511 537, 588-610 multiplying and sales tax, calculating, real-life applications 107-109 for, 614-615 dividing, 527-532 solving by completing perfect cubes, 511-512 sample, 331 the square, 561-572 perfect squares, sample space, 367-368 solving by factoring, sampling, 330-332 541-549 508-509 scale drawings, 79 solving by graphing, simplifying, 515-518 scatter plots, 353-357 611-618 square roots, 506-508 scientific notation, 149-154 solving by taking radicand, 521-524 set, 335 square roots, random probability, 364 simple interest, 123-130 553-560 random sample, 331 simplifying solving with the range, 252, 339, 396-406 Quadratic Formula, rate, 83 cube roots, 510-511, 573-578 rational numbers, 3, 5, 517-518 508-509, 511-512 rationalizing the denominator, 529-530 625

exponents and, 437-444 perfect squares, order of operations monomials and, 445-450 508-509 and, 17-22 multiplying and dividing simplifying, 515-516 of polynomials, polynomials and, solving quadratic 428-434 453-460 radicals, 515-518 equations by taking, of positive and ratios, 78 553-560 negative fractions, square roots, 507 squares formula, 53-56 slope difference of two, definition of, 263 495-496, 501-502 of positive and description of, standard form of negative whole 263-265 equation, 294 numbers, 33-36 finding, 268-272 standard form of negative, 267, 272 expression, 158-160 of radicals, 521-524 overview of, 263-276 standard notation, sum of two cubes formula, positive, 267, 271 scientific notation undefined, 267 and, 149-154 499-502 zero, 267 statistical question, symbols slope-intercept form, 329 277-283, 295, statistics inequality symbols, 315-316, 319 definition of, 326 196, 200, 212 solutions introduction to, checking, 179 326-332 infinity symbol, 215 definition of, 176-177 substitution radical sign, 506 of inequalities, 198-201 evaluating algebraic symmetry, axis of, for quadratic equations, expressions and, 536 163-168 599-600 of system of linear solving systems of system of linear equations equations, 226 linear equations by, square, solving quadratic 225-234 solving by elimination, equations by substitution method, 237-244 completing, 561-572 227-234 Square Root Property, subtraction solving by graphing, 554-557, 561 of decimals, 59-60, 299-306 square roots 62-63 description of, Distributive Property solving by substitution, 506-508 of Multiplication over, 225-234 13, 14 See also linear equations t tables ratios and, 135-137 two-way, 343-345 of values, graphing lines from, 255-257 626

terminating decimals, 6 maximum, 337, 597-598 whole numbers terms, 155-156 minimum, 337, 597-598 adding positive and time, for loan, 124 variability, 329 negative, 24-29 tree diagram, 367-368 variable(s) definition of, 2 trinomials definition of, 156 multiplying and dependent, 177-179, dividing positive and definition of, 156, 428 negative, 37-39 factoring when a = 1, 396-397 in number system, 5 with exponents, 157-158 subtracting positive 481-490 independent, 177-179, and negative, 33-36 factoring when a ≠ 1, 396-397 x 491-492 isolating, 185-194 x-axis, 248, 277-278 perfect square trinomial solving equations with x-coordinates, 248-252 x-intercepts, 277-283, 288 formula, 497-498 one, 185-191 two-way tables, 343-345 solving inequalities with Y y-axis, 248, 277-278 u one, 195-201 y-coordinates, 248-254 undefined slope, 267 variation y-intercepts, 277-283, 288 union (or), 210-211, 549 unit price, 85-86 constant of, 93 Z unit rate, 83-86 measures of, 339-340 zero (x-intercept), unknown quantity, finding, vertex form, 593-596 vertex of parabola, 601-602 91-92 zero slope, 267 590-596 Zero-Product Principle, v vertical line test (VLT), value(s) 541-548 400-403 absolute, 27-29, 61-63 graphing a line from w whole, finding when given table of, 255-257 percent, 103-104 627

Goodbye, Algebra. . . Hello, GEOMETRY! This BIG FAT NOTEBOOK covers everything you need to know during a year of GEOMETRY class, breaking down one big fat subject into accessible units, from the basics like points, lines, planes, and angles to the beginning of trigonometry. P.S. Are you one of those brainiacs taking Algebra in middle school? Don’t forget to ace the rest of your classes with the original BIG FAT NOTEBOOKS series: