Quadratic Functions Algebra III, Sec. 2.1 Objective You will learn how to sketch and analyze graph of functions.

Important Vocabulary  Axis – the line of symmetry for a parabola  Vertex – the point where the the axis intersects the parabola

The Graph of a Quadratic Fn  A polynomial function of x with degree n is  A quadratic function is

The Graph of a Quadratic Fn (cont.) A quadratic function is a polynomial function of _____________ degree. The graph of a quadratic function is a special “U” shaped curve called a _____________. second parabola

The Graph of a Quadratic Fn (cont.) If the leading coefficient of a quadratic function is positive, the graph of the function opens ________ and the vertex of the parabola is the ____________ y-value on the graph. up minimum

The Graph of a Quadratic Fn (cont.) If the leading coefficient of a quadratic function is negative, the graph of the function opens ________ and the vertex of the parabola is the ____________ y-value on the graph. down maximum

The Graph of a Quadratic Fn (cont.) The absolute value of the leading coefficient determines ____________________________________. If |a| is small, __________________________________ _______________________________________________. how wide the parabola opens the parabola opens wider than when |a| is large

Example 1 Sketch and compare the graphs of the quadratic functions. a) Reflects over the x-axis Vertical stretch by 3/2

Example 1 Sketch and compare the graphs of the quadratic functions. b) Vertical shrink by 5/6

Standard Form of a Quadratic Fn The standard form of a quadratic function is ________________________________. The axis of the associated parabola is ___________ and the vertex is ____________. a ≠ 0 x = h (h, k)

Standard Form of a Quadratic Fn To write a quadratic function in standard form… Use completing the square add and subtract the square of half the coefficient of x

Example 2 Sketch the graph of f(x). Identify the vertex and axis. Write original function. Add & subtract (b/2) 2 within parentheses. Regroup terms. Simplify. Write in standard form.

Example 2 Sketch the graph of f(x). Identify the vertex and axis. a = 1 h = 5 k = 0 Vertex: (5, 0) Axis: x = 5

Standard Form of a Quadratic Fn To find the x-intercepts of the graph… You must solve the quadratic equation.

Example 3 Sketch the graph of f(x). Identify the vertex and x- intercepts. Write original function. Factor -1 out of x terms. Add & subtract (b/2) 2 within parentheses. Regroup terms. Simplify. Write in standard form.

Example 3 Sketch the graph of f(x). Identify the vertex and x- intercepts. a = -1 h = -2 k = 25 Vertex: (-2, 25) Axis: x = -2

Example 3 Sketch the graph of f(x). Identify the vertex and x- intercepts. Set original function =0. Factor out -1. Factor. Set factors =0. Solve.

Example (on your handout) Sketch the graph of f(x). Identify the vertex and x- intercepts. Write original function. Add & subtract (b/2) 2 within parentheses. Regroup terms. Simplify. Write in standard form.

Example (on your handout) Sketch the graph of f(x). Identify the vertex and x- intercepts. a = 1 h = -1 k = -9 Vertex: (-1, -9) Axis: x = -1

Example (on your handout) Sketch the graph of f(x). Identify the vertex and x- intercepts. Set original function =0. Factor. Set factors =0. Solve.

Applications of Quadratic Fns For a quadratic function in the form, the x-coordinate of the vertex is given as ___________ & the y-coordinate of the vertex is given as ________.

Example 5 The height y (in feet) of a ball thrown by a child is given by, where x is the horizontal distance (in feet) from where the ball is thrown. How high is the ball when it is at its maximum height? a = - 1/8 b = 1 c = 4 Maximum height = vertex

Example (on your handout) Find the vertex of the parabola defined by