Key Components for Graphing a Quadratic Function.

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Presentation transcript:

Key Components for Graphing a Quadratic Function

Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a ≠ 0)

Quadratic Functions

Parabolas Parabola: The graph of a quadratic function If a > 0, the parabola opens up If a < 0, the parabola opens down Vertex: highest / lowest point of a parabola. We called these coordinates. If the parabola opens up, is the vertex a maximum or minimum?

Parabolas Axis of symmetry: Vertical line passing through the vertex

Parabolas Locations of vertex and axis of symmetry: You will need the formula This gives you the x-coordinate of the vertex AND the axis of symmetry. How do you find the y-coordinate of the vertex?

Parabolas Example. For the parabola defined by f(x) = 2x 2 { 3x + 2 (a) Problem: Without graphing, locate the vertex. Answer: (b) Problem: Does the parabola open up or down? Answer:

x-intercepts of a Parabola For a quadratic function f(x) = ax 2 + bx + c: Discriminant is b 2 { 4ac. Number of x-intercepts depends on the discriminant. Positive discriminant: Two x-intercepts Negative discriminant: Zero x-intercepts Zero discriminant: One x-intercept (Vertex lies on x-axis)

x-intercepts of a Parabola Positive discriminant Zero discriminant Negative discriminant

Graphing Quadratic Functions Example. For the function f(x) = 2x 2 + 8x + 4 (a) Problem: Find the vertex Answer: (b) Problem: Find the x- and y-intercepts. Answer:

Graphing Quadratic Functions Graphing a rough sketch Determine the vertex Determine the axis of symmetry Determine the y-intercept (How?) Find the discriminant b 2 { 4ac. Solve for the x-intercepts (How?)

Graphing Quadratic Functions Example. For the quadratic function f(x) = 3x 2 { 12x + 7 (a) Problem: Determine whether f has a maximum or minimum value, then find it. Answer: