Quadratic Quadratic Functions and Functions and Their GraphsTheir Graphs

More in Sec. 2.1bMore in Sec. 2.1b

What are they???

Quadratic Function – a polynomial function ofdegree 2

Recall the basic squaring function? 2f x x Any quadratic function can be obtained via aAny quadratic function can be obtained via a sequence of transformations of this basicsequence of transformations of this basic function…………observe............function…………observe............

Quick Examples

Describe how to transform the basic squaring function into thegraph of the given function. Sketch its graph by hand.

213

2g x x

Vertical shrink by 1/2,Vertical shrink by 1/2,reflection across x-axis,reflection across x-axis,translation up 3 unitstranslation up 3 units

Quick Examples

Describe how to transform the basic squaring function into thegraph of the given function. Sketch its graph by hand.

23 2 1h x x

Translation left 2 units,Translation left 2 units,vertical stretch by 3,vertical stretch by 3,

translation down 1 unittranslation down 1 unit

More Generalizations…

Consider the graph of 2f x ax0a If , the parabola

opens upward0a If , the parabola

opens downward

Axis of Symmetry (axis for short) – line of symmetry

Vertex – point where the parabola intersects the axis

2f x ax bx c 0a

Definition: Vertex Form of a Quadratic Function

Any quadratic function , , can bewritten in the vertex form

2f x a x h k The graph of f is a parabola with vertex (h, k ) and axis x = h,where and . If a > 0, the parabolaopens upward, and if a < 0, it opens downward.

2k c ah / 2h b a

(Standard Quadratic Form)(Standard Quadratic Form)

Guided Practice

Find the vertex and axis of the graph of the given functions.

27 2 5k x x Vertex: (–2, 5)Vertex: (–2, 5)Axis: x = –2Axis: x = –2

2

1 31

4 2g x x

Vertex: (3/2, –1)Vertex: (3/2, –1)

Axis: x = 3/2Axis: x = 3/2

Guided PracticeUse vertex form of a quadratic function to find the vertex andaxis of the given function. Rewrite the equation in vertex form.

26 3 5f x x x Standard form: 23 6 5f x x x

So, a = –3, b = 6, and c = –5

Coordinates of the vertex:

6

12 3

2

bh

a 2 1k f h f

Guided PracticeUse vertex form of a quadratic function to find the vertex andaxis of the given function. Rewrite the equation in vertex form.

26 3 5f x x x Vertex: 1, 2

Vertex form of f : 23 1 2f x x

Axis: 1x

How about a graph to support these answers?How about a graph to support these answers?

First, let’s make sure we remember how to complete the square…

3 7x

Solve by completing the square:

2 6 2x x

2 6 2 0x x

2 6 9 2 9x x

3 7x 23 7x

Get x terms by themselvesGet x terms by themselves

Complete the square!!!Complete the square!!!

FactorFactor

Take square root of both sidesTake square root of both sides

Solve for xSolve for x

We can complete a similar process when changing forms of quadratics:

23 2 1x

Use completing the square to describe the graph of thegiven function. Support your answer graphically.

23 12 11f x x x

The graph of The graph of ff is a upward-opening parabola is a upward-opening parabolawith vertex (–2, –1), axiswith vertex (–2, –1), axis x x = –2, and intersects= –2, and intersectsthe the xx-axis at about –2.577 and –1.423.-axis at about –2.577 and –1.423.

23 4 11x x

23 4 4 11 12x x

Characterizing the Nature of a Quadratic Function

2f x ax bx c

Point of View Characterization

Verbal Polynomial of degree 2

0a Algebraic or

2f x a x h k Graphical Parabola with vertex (h, k), axis x = h;

opens upward if a > 0, opens downwardif a < 0; initial value = y-int = f(0) = c;x-intercepts: 2 4

2

b b ac

a

Guided Practice

25 77

52 4

x

Use completing the square to describe the graph of thegiven function. Support your answer graphically.

25 25 12f x x x

Vertex: (5/2, –77/4), Axis:Vertex: (5/2, –77/4), Axis: x x = 5/2, Opens= 5/2, Opensupward, intersects the upward, intersects the xx-axis at about 0.538-axis at about 0.538and 4.462, Vertically stretched by 5.and 4.462, Vertically stretched by 5.

Guided PracticeWrite an equation for the parabola shown, using the factthat one of the given points is the vertex.

23 2y a x

(3, –2)

(6, 1)

Plug in (3, –2) for (h, k):

21 6 3 2a

Plug in (6, 1) for (x, y), solve for a:1

3a

213 2

3f x x

Check with aCheck with acalculator graph!!!calculator graph!!!

Guided PracticeWrite an equation for the parabola shown, using the factthat one of the given points is the vertex.

21 5y a x

(2,–13)

(–1, 5)Plug in (–1, 5) for (h, k):

213 2 1 5a

Plug in (2,–13) for (x, y), solve for a:

2a

22 1 5f x x Check with aCheck with a

calculator graph!!!calculator graph!!!

Guided PracticeWrite an equation for the quadratic function whose graphcontains the vertex (–2, –5) and the point (–4, –27).

Check with aCheck with acalculator graph!!!calculator graph!!!

Plug in the vertex: 22 5f x a x

Plug in the point: 24 4 2 5 27f a 11 2a

2112 5

2f x x