![]() ![]() Check Substitute each value into original equation. x = –1 or x = 6 Solve each equation.įind the zeros of the function by factoring. ![]() x + 1 = 0 or x – 6 = 0 Apply the Zero Product Property. (x + 1)(x – 6) = 0 Factor: Find factors of –6 that add to –5. f(x)= x2 – 5x – 6 x2 – 5x – 6 = 0 Set the function equal to 0. x = 0 or x = –6 Solve each equation.Ģ1 Check Check algebraically and by graphing.Įxample 2B Continued Check Check algebraically and by graphing. 3x = 0 or x + 6 = 0 Apply the Zero Product Property. g(x) = 3x2 + 18x 3x2 + 18x = 0 Set the function to equal to 0. x= –2 or x = 6 Solve each equation.ġ9 Find the zeros of the function by factoring.Įxample 2A Continued Find the zeros of the function by factoring. x + 2 = 0 or x – 6 = 0 Apply the Zero Product Property. (x + 2)(x – 6) = 0 Factor: Find factors of –12 that add to –4. f(x) = x2 – 4x – 12 x2 – 4x – 12 = 0 Set the function equal to 0. Reading Mathġ8 Example 2A: Finding Zeros by Factoringįind the zeros of the function by factoring. The roots of an equation are the values of the variable that make the equation true.ġ7 You can find the roots of some quadratic equations by factoring and applying the Zero Product Property. The solution to a quadratic equation of the form ax2 + bx + c = 0 are roots. The solutions to the related equation x2 + 2x – 3 = 0 represent the zeros of the function. For example, to find the zeros of f(x)= x2 + 2x – 3, you can set the function equal to zero. You can also find zeros by using algebra. These are the zeros of the function.ġ6 You can also find zeros by using algebra Both the table and the graph show that y = 0 at x = –3 and x = 1. Enter y = –x2 – 2x + 3 into a graphing calculator. x –3 –2 –1 1 f(x) 3 4 (–3, 0) (1, 0) The table and the graph indicate that the zeros are –3 and 1.įind the zeros of f(x) = –x2 – 2x + 3 by using a graph and table. Use symmetry and a table of values to find additional points. The x-coordinate of the vertex is Find the vertex:įind the zeros of g(x) = –x2 – 2x + 3 by using a graph and table. Method 1 Graph the function g(x) = –x2 – 2x + 3. These are the zeros of the function.ġ2 Find the zeros of g(x) = –x2 – 2x + 3 by using a graph and a table.Ĭheck It Out! Example 1 Find the zeros of g(x) = –x2 – 2x + 3 by using a graph and a table. Both the table and the graph show that y = 0 at x = 2 and x = 4. Enter y = x2 – 6x + 8 into a graphing calculator. x 1 2 3 4 5 f(x) –1 (4, 0) (2, 0) The table and the graph indicate that the zeros are 2 and 4.ġ1 Example 1 Continued Find the zeros of f(x) = x2 – 6x + 8 by using a graph and table. The x-coordinate of the vertex is Find the vertex:ĩ Find the zeros of f(x) = x2 – 6x + 8 by using a graph and table.Įxample 1 Continued Find the zeros of f(x) = x2 – 6x + 8 by using a graph and table. Method 1 Graph the function f(x) = x2 – 6x The graph opens upward because a > 0. Helpful HintĨ Example 1: Finding Zeros by Using a Graph or Tableįind the zeros of f(x) = x2 – 6x + 8 by using a graph and table. These zeros are always symmetric about the axis of symmetry.ħ 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. Unlike linear functions, which have no more than one zero, quadratic functions can have two zeros, as shown at right. The zeros of a function are the x-intercepts. When the ball hits the ground, the value of the function is zero.Ħ A zero of a function is a value of the input x that makes the output f(x) equal zero. In this situation, the value of the function represents the height of the soccer ball. x2 – 49 (x – 7)(x + 7)ģ Objectives Solve quadratic equations by graphing or factoring.ĭetermine a quadratic function from its roots.Ĥ Vocabulary zero of a function root of an equation binomial trinomialĥ When a soccer ball is kicked into the air, how long will the ball take to hit the ground? The height h in feet of the ball after t seconds can be modeled by the quadratic function h(t) = –16t2 + 32t. 1 Solving quadratic equations by graphing and factoringĢ Warm Up Find the x-intercept of each function. ![]()
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