Algebra 1 Section 12.9

The y-intercept of a Function Since the y-intercept always has an x value of zero, the y-intercept can be found by letting x = 0 and solving for y = f(0).

Example 1 g(x) = 2x2 – 3x + 5 g(0) = 2(0)2 – 3(0) + 5 = 5 y-intercept: (0, 5) h(x) = (x – 1)2 + 2 h(0) = (0 – 1)2 + 2 = 3 y-intercept: (0, 3)

Zeroes of a Function The x-intercepts of a graph always have y values of zero. Therefore, the x-intercepts can be found by letting y = 0 or f(x) = 0 and solving for x.

Definition A zero (or root ) of a function is a value of x at which the graph of the function crosses the x- axis. It is the x-coordinate of an x-intercept of the graph.

Example 2a Find the zero(s) of r(x) = x2 – 4x – 5 x2 – 4x – 5 = 0 x – 5 = 0 or x + 1 = 0 x = 5, -1

Example 2b s(x) = -x2 + 4x – 4 -x2 + 4x – 4 = 0 -1(x2 – 4x + 4) = 0 x – 2 = 0 or x – 2 = 0 x = 2 or x = 2

Double Root In Example 2b, the single root is often called a double root since two factors produce the same root. In this case, the parabola’s vertex is located at that x- intercept.

Example 3a Find the zero(s) of p(x) = (x – 1)2 – 6 (x – 1)2 – 6 = 0

Example 3a x = 1 ± 6 x = 1 + 6 ≈ 3.45 x = 1 – 6 ≈ -1.45

Example 3b Find the zero(s) of q(x) = x2 – x + 4 x2 – x + 4 = 0 1 ± -15 2 There are no real zeros.

No Real Zeros In Example 3b, there are no x- intercepts. There are no x-intercepts or zeros when the discriminant is negative.

Example 4 f(x) = 3x2 + 8x – 3 The y-intercept is found by finding f(0). f(0) = 3(0)2 + 8(0) – 3 = -3 y-intercept: (0, -3)

Example 4 f(x) = 3x2 + 8x – 3 The zeros are found by setting the function equal to zero. 3x2 + 8x – 3 = 0 (3x – 1)(x + 3) = 0 3x – 1 = 0 or x + 3 = 0

Example 4 f(x) = 3x2 + 8x – 3 The zeros are found by setting the function equal to zero. 3x – 1 = 0 or x + 3 = 0 x = ⅓ or x = -3 The zeros are ⅓ and -3.

Example 4 f(x) = 3x2 + 8x – 3 The vertex is (h, k). Use the formula to find h. a = 3, b = 8, c = -3 h = - b 2a = - 8 2(3) = - 4 3 or -1⅓

Example 4 f(x) = 3x2 + 8x – 3 To find k, use k = f(h). k = f(- ) = 3(- )2 + 8(- ) – 3 4 3 = - 25 3 or -8⅓ The vertex is (- , - ). 4 3 25 The vertex is (-1⅓, -8⅓).

Homework: pp. 524-526