Warm Up 1. y = 2x – 3 2. 3. y = 3x + 6 4. y = –3x2 + x – 2, when x = 2 Find the x-intercept of each linear function. 1. y = 2x – 3 2. 3. y = 3x + 6 Evaluate each quadratic function for the given input values. 4. y = –3x2 + x – 2, when x = 2 5. y = x2 + 2x + 3, when x = –1 –2 –12 2

Factor each polynomial. 3x3 + 2x2 – 10 The polynomial cannot be factored further. –18y3 – 7y2 8x4 + 4x3 – 2x2

Objectives Find the zeros of a quadratic function from its graph. Find the axis of symmetry and the vertex of a parabola.

Recall that an x-intercept of a function is a value of x when y = 0 Recall that an x-intercept of a function is a value of x when y = 0. A zero of a function is an x-value that makes the function equal to 0. So a zero of a function is the same as an x-intercept of a function. Since a graph intersects the x-axis at the point or points containing an x-intercept, these intersections are also at the zeros of the function. A quadratic function may have one, two, or no zeros.

Find the zeros of the quadratic function from its graph Find the zeros of the quadratic function from its graph. Check your answer. y = x2 – 2x – 3 y = (–1)2 – 2(–1) – 3 = 1 + 2 – 3 = 0 y = 32 –2(3) – 3 = 9 – 6 – 3 = 0 y = x2 – 2x – 3 Check  The zeros appear to be –1 and 3.

Find the zeros of the quadratic function from its graph Find the zeros of the quadratic function from its graph. Check your answer. y = x2 + 8x + 16 Check y = x2 + 8x + 16 y = (–4)2 + 8(–4) + 16 = 16 – 32 + 16 = 0  The zero appears to be –4.

Notice that if a parabola has only one zero, the zero is the x-coordinate of the vertex. Helpful Hint

Find the zeros of the quadratic function from its graph Find the zeros of the quadratic function from its graph. Check your answer. y = –2x2 – 2 The graph does not cross the x-axis, so there are no zeros of this function.

Find the zeros of the quadratic function from its graph Find the zeros of the quadratic function from its graph. Check your answer. y = x2 – 6x + 9 y = (3)2 – 6(3) + 9 = 9 – 18 + 9 = 0 y = x2 – 6x + 9 Check  The zero appears to be 3.

A vertical line that divides a parabola into two symmetrical halves is the axis of symmetry. The axis of symmetry always passes through the vertex of the parabola. You can use the zeros to find the axis of symmetry.

Find the axis of symmetry of each parabola. (–1, 0) The axis of symmetry is x = –1. B. The axis of symmetry is x = 2.5.

Find the axis of symmetry of each parabola. (–3, 0) The axis of symmetry is x = –3. b. The axis of symmetry is x = 1.

If a function has no zeros or they are difficult to identify from a graph, you can use a formula to find the axis of symmetry. The formula works for all quadratic functions.

Find the axis of symmetry of the graph of y = –3x2 + 10x + 9. Step 2. Use the formula. a = –3, b = 10 The axis of symmetry is

Find the axis of symmetry of the graph of y = 2x2 + x + 3. Step 2. Use the formula. a = 2, b = 1 The axis of symmetry is .

Once you have found the axis of symmetry, you can use it to identify the vertex.

Find the vertex. y = 0.25x2 + 2x + 3 = 0.25(–4)2 + 2(–4) + 3 = –1 Step 3 Write the ordered pair. (–4, –1)

Find the vertex. a = –3, b = 10 y = –3x2 + 6x – 7 The x-coordinate of the vertex is 1. = –3(1)2 + 6(1) – 7 = –3 + 6 – 7 = –4 The vertex is (1, –4).

The graph of f(x) = –0. 06x2 + 0. 6x + 10 The graph of f(x) = –0.06x2 + 0.6x + 10.26 can be used to model the height in meters of an arch support for a bridge, where the x-axis represents the water level and x represents the distance in meters from where the arch support enters the water. Can a sailboat that is 14 meters tall pass under the bridge? Explain. The vertex represents the highest point of the arch support. a = – 0.06, b = 0.6 = –0.06(5)2 + 0.6(5) + 10.26 = 11.76 The sailboat cannot pass under the bridge.