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Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring 5-3 Solving Quadratic Equations by Graphing and Factoring Holt Algebra 2 Warm.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Warm Up Find the x-intercept of each function.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Solve quadratic equations by graphing or factoring. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring When a soccer ball is kicked into the air, how long will the ball take to hit the ground? Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring A zero of a function is a value of the input x that makes the output f(x) equal zero.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Recall that for the graph of a quadratic function, any pair of points with the same y-value are symmetric about the axis of symmetry. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the zeros of f(x) = x 2 – 6x + 8 by using a graph and table.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring x12345 f(x)f(x)30–1–103 Plot the vertex and the y-intercept.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Method 1 Graph the function g(x) = –x 2 – 2x + 3. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the zeros of g(x) = –x 2 – 2x + 3 by using a graph and table. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Plot the vertex and the y-intercept. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the zeros of f(x) = –x 2 – 2x + 3 by using a graph and table. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring You can also find zeros by using algebra.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring You can find the roots of some quadratic equations by factoring and applying the Zero Product Property.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the zeros of the function by factoring.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Example 2B Continued Check Check algebraically and by graphing.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Any object that is thrown or launched into the air, such as a baseball, basketball, or soccer ball, is a projectile.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Example 3 Continued The ball will hit the ground when its height is zero. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Check The graph of the function h(t) = –16t 2 + 80t shows its zeros at 0 and 5. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Quadratic expressions can have one, two or three terms, such as –16t 2, –16t 2 + 25t, or –16t 2 + 25t + 2.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the roots of the equation by factoring.

Factor by Grouping Example of a polynomial: (10x 2 + 5x – 2x – 1) Change to (10x 2 +5x)(-2x -1) Take GCF of both binomials.

Student will be able to factor Quadratic Trinomials of the form Leading coefficient not = 1 Leading coefficient not = 1. Factoring Polynomials Grouping, Trinomials, Binomials, GCF,Quadratic form & Solving Equations. Factoring Polynomials Factoring a polynomial means expressing it as a product of other polynomials.

Solve Notice that if you take ? of the middle number and square it, you get the last number. The height h in feet of the ball after t seconds can be modeled by the quadratic function h(t) = –16t 2 + 32t. Example 1: Finding Zeros by Using a Graph or Table Method 1 Graph the function f(x) = x 2 – 6x + 8. Example 1 Continued Find f(3): f(x) = x 2 – 6x + 8 f(3) = (3) 2 – 6(3) + 8 f(3) = 9 – 18 + 8 f(3) = –1 Substitute 3 for x. Find g(1): g(x) = –x 2 – 2x + 3 g(–1) = –(–1) 2 – 2(–1) + 3 g(–1) = –1 + 2 + 3 g(–1) = 4 Substitute –1 for x. For example, to find the zeros of f(x)= x 2 + 2x – 3, you can set the function equal to zero. Example 2A: Finding Zeros by Factoring f(x) = x 2 – 4x – 12 x 2 – 4x – 12 = 0 (x + 2)(x – 6) = 0 x + 2 = 0 or x – 6 = 0 x= –2 or x = 6 Set the function equal to 0.

Example 2A Continued Check (–2) 2 – 4(–2) – 12 4 + 8 – 12 0 x 2 – 4x – 12 = 0 0 0 0 (6) 2 – 4(6) – 12 36 – 24 – 12 0 x 2 – 4x – 12 = 0 0 0 0 Substitute each value into original equation. Example 2B: Finding Zeros by Factoring g(x) = 3x 2 + 18x 3x 2 + 18x = 0 3x(x+6) = 0 3x = 0 or x + 6 = 0 x = 0 or x = –6 Set the function to equal to 0. Check (–1) 2 – 5(–1) – 6 1 + 5 – 6 0 x 2 – 5x – 6 = 0 0 0 0 (6) 2 – 5(6) – 6 36 – 30 – 6 0 x 2 – 5x – 6 = 0 0 0 0 Substitute each value into original equation.

Check (0) 2 – 8(0) 0 – 0 0 x 2 – 8x = 0 0 0 0 (8) 2 – 8(8) 64 – 64 0 x 2 – 8x = 0 0 0 0 Substitute each value into original equation. The general function that approximates the height h in feet of a projectile on Earth after t seconds is given. Example 3 Continued Check The graph of the function h(t) = –16t 2 + 48t shows its zeros at 0 and 3. Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note- taking materials. Exponent Rules a m · a n = a m+n x 2 · x 3 = x 5 a m = a m-n x 7 = x 2 a n x 5 a -n = 1 x -3 = 1 a n x 3 (a m ) n = a mn (x.

Monomial Is an expression that is a number, variable, or a product of a number and one or more variables.

Unlike linear functions, which have no more than one zero, quadratic functions can have two zeros, as shown at right.

The solutions to the related equation x 2 + 2x – 3 = 0 represent the zeros of the function. Note that this model has limitations because it does not account for air resistance, wind, and other real-world factors. Example 3: Sports Application h(t) = –16t 2 + v 0 t + h 0 h(t) = –16t 2 + 80t + 0 Write the general projectile function. Check: FOIL the two binomials and if the product is the beginning polynomial, the two binomials are correct.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Warm Up Find the x-intercept of each function.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Solve quadratic equations by graphing or factoring. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring When a soccer ball is kicked into the air, how long will the ball take to hit the ground? Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring A zero of a function is a value of the input x that makes the output f(x) equal zero.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Recall that for the graph of a quadratic function, any pair of points with the same y-value are symmetric about the axis of symmetry. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the zeros of f(x) = x 2 – 6x + 8 by using a graph and table.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring x12345 f(x)f(x)30–1–103 Plot the vertex and the y-intercept.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Method 1 Graph the function g(x) = –x 2 – 2x + 3. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the zeros of g(x) = –x 2 – 2x + 3 by using a graph and table. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Plot the vertex and the y-intercept. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the zeros of f(x) = –x 2 – 2x + 3 by using a graph and table. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring You can also find zeros by using algebra.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring You can find the roots of some quadratic equations by factoring and applying the Zero Product Property.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the zeros of the function by factoring.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Example 2B Continued Check Check algebraically and by graphing.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Any object that is thrown or launched into the air, such as a baseball, basketball, or soccer ball, is a projectile.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Example 3 Continued The ball will hit the ground when its height is zero. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Check The graph of the function h(t) = –16t 2 + 80t shows its zeros at 0 and 5. Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Quadratic expressions can have one, two or three terms, such as –16t 2, –16t 2 + 25t, or –16t 2 + 25t + 2.

Holt Algebra 2 5-3 Solving Quadratic Equations by Graphing and Factoring Find the roots of the equation by factoring.

Factor by Grouping Example of a polynomial: (10x 2 + 5x – 2x – 1) Change to (10x 2 +5x)(-2x -1) Take GCF of both binomials.

Student will be able to factor Quadratic Trinomials of the form Leading coefficient not = 1 Leading coefficient not = 1. Factoring Polynomials Grouping, Trinomials, Binomials, GCF,Quadratic form & Solving Equations. Factoring Polynomials Factoring a polynomial means expressing it as a product of other polynomials.

Solve Notice that if you take ? of the middle number and square it, you get the last number. The height h in feet of the ball after t seconds can be modeled by the quadratic function h(t) = –16t 2 + 32t. Example 1: Finding Zeros by Using a Graph or Table Method 1 Graph the function f(x) = x 2 – 6x + 8. Example 1 Continued Find f(3): f(x) = x 2 – 6x + 8 f(3) = (3) 2 – 6(3) + 8 f(3) = 9 – 18 + 8 f(3) = –1 Substitute 3 for x. Find g(1): g(x) = –x 2 – 2x + 3 g(–1) = –(–1) 2 – 2(–1) + 3 g(–1) = –1 + 2 + 3 g(–1) = 4 Substitute –1 for x. For example, to find the zeros of f(x)= x 2 + 2x – 3, you can set the function equal to zero. Example 2A: Finding Zeros by Factoring f(x) = x 2 – 4x – 12 x 2 – 4x – 12 = 0 (x + 2)(x – 6) = 0 x + 2 = 0 or x – 6 = 0 x= –2 or x = 6 Set the function equal to 0.

Example 2A Continued Check (–2) 2 – 4(–2) – 12 4 + 8 – 12 0 x 2 – 4x – 12 = 0 0 0 0 (6) 2 – 4(6) – 12 36 – 24 – 12 0 x 2 – 4x – 12 = 0 0 0 0 Substitute each value into original equation. Example 2B: Finding Zeros by Factoring g(x) = 3x 2 + 18x 3x 2 + 18x = 0 3x(x+6) = 0 3x = 0 or x + 6 = 0 x = 0 or x = –6 Set the function to equal to 0. Check (–1) 2 – 5(–1) – 6 1 + 5 – 6 0 x 2 – 5x – 6 = 0 0 0 0 (6) 2 – 5(6) – 6 36 – 30 – 6 0 x 2 – 5x – 6 = 0 0 0 0 Substitute each value into original equation.

Check (0) 2 – 8(0) 0 – 0 0 x 2 – 8x = 0 0 0 0 (8) 2 – 8(8) 64 – 64 0 x 2 – 8x = 0 0 0 0 Substitute each value into original equation. The general function that approximates the height h in feet of a projectile on Earth after t seconds is given. Example 3 Continued Check The graph of the function h(t) = –16t 2 + 48t shows its zeros at 0 and 3. Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note- taking materials. Exponent Rules a m · a n = a m+n x 2 · x 3 = x 5 a m = a m-n x 7 = x 2 a n x 5 a -n = 1 x -3 = 1 a n x 3 (a m ) n = a mn (x.

Monomial Is an expression that is a number, variable, or a product of a number and one or more variables.

Unlike linear functions, which have no more than one zero, quadratic functions can have two zeros, as shown at right.

The solutions to the related equation x 2 + 2x – 3 = 0 represent the zeros of the function. Note that this model has limitations because it does not account for air resistance, wind, and other real-world factors. Example 3: Sports Application h(t) = –16t 2 + v 0 t + h 0 h(t) = –16t 2 + 80t + 0 Write the general projectile function. Check: FOIL the two binomials and if the product is the beginning polynomial, the two binomials are correct.

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