section 2.4 the graph of a quadratic function the graph of a quadratic function

SECTION 2.4SECTION 2.4

THE GRAPH OF A QUADRATIC THE GRAPH OF A QUADRATIC FUNCTIONFUNCTION

GRAPHS OF QUADRATIC FUNCTIONS


As we’ve already seen, f(x) = xAs we’ve already seen, f(x) = x2 2 graphs into a PARABOLA.graphs into a PARABOLA.

This is the simplest quadratic This is the simplest quadratic function we can think of. We will function we can think of. We will use this one as a model by which use this one as a model by which to compare all other quadratic to compare all other quadratic functions we will examine.functions we will examine.

VERTEX OF A PARABOLA

VERTEX OF A PARABOLA

All parabolas have a VERTEX, the All parabolas have a VERTEX, the lowest or highest point on the lowest or highest point on the graph (depending upon whether it graph (depending upon whether it opens up or down.opens up or down.

AXIS OF SYMMETRYAXIS OF SYMMETRY

All parabolas have an AXIS OF All parabolas have an AXIS OF SYMMETRY, an imaginary line SYMMETRY, an imaginary line which goes through the vertex which goes through the vertex and about which the parabola is and about which the parabola is symmetric.symmetric.

HOW PARABOLAS DIFFER

HOW PARABOLAS DIFFER

Some parabolas open up and Some parabolas open up and some open down.some open down.

Parabolas will all have a different Parabolas will all have a different vertex and a different axis of vertex and a different axis of symmetry.symmetry.

Some parabolas will be wide and Some parabolas will be wide and some will be narrow.some will be narrow.



The standard form of a quadratic The standard form of a quadratic function is:function is:

f(x) = axf(x) = ax22 + bx + c + bx + c

The position, width, and The position, width, and orientation of a particular orientation of a particular parabola will depend upon the parabola will depend upon the values of a, b, and c.values of a, b, and c.



Compare f(x) = xCompare f(x) = x22 to the to the following:following:

f(x) = 2xf(x) = 2x22 f(x) = .5x f(x) = .5x2 2 f(x) = f(x) = -.5x-.5x22

If a > 0, then the parabola opens If a > 0, then the parabola opens upup

If a < 0, then the parabola opens If a < 0, then the parabola opens downdown



Now compare f(x) = xNow compare f(x) = x22 to the to the following:following:

f(x) = x f(x) = x 22 + 3+ 3 f(x) = x f(x) = x 22 - 2 - 2

Vertical shift upVertical shift up Vertical shift Vertical shift downdown



Now compare f(x) = xNow compare f(x) = x22 to the to the following:following:

f(x) = (x + 2)f(x) = (x + 2)22 f(x) = (x – 3)f(x) = (x – 3)22

Horizontal shift Horizontal shift to the leftto the left

Horizontal shift Horizontal shift to the rightto the right



When the standard form of a When the standard form of a quadratic function f(x) = axquadratic function f(x) = ax22 + bx + c + bx + c is written in the form:is written in the form:

a(x - h) a(x - h) 22 + k + k

We can tell by horizontal and vertical We can tell by horizontal and vertical shifting of the parabola where the shifting of the parabola where the vertex will be.vertex will be.

The parabola will be shifted h units The parabola will be shifted h units horizontally and k units vertically.horizontally and k units vertically.



Thus, a quadratic function written in Thus, a quadratic function written in the form the form

a(x - h) a(x - h) 22 + k + k

will have a vertex at the point (h,k).will have a vertex at the point (h,k).

The value of “a” will determine The value of “a” will determine whether the parabola opens up or whether the parabola opens up or down (positive or negative) and down (positive or negative) and whether the parabola is narrow or whether the parabola is narrow or wide.wide.



a(x - h) a(x - h) 22 + k + k

Vertex (highest or lowest point): Vertex (highest or lowest point): (h,k)(h,k)

If a > 0, then the parabola opens If a > 0, then the parabola opens upup

If a < 0, then the parabola opens If a < 0, then the parabola opens downdown



Axis of SymmetryAxis of Symmetry

The vertical line about which the The vertical line about which the graph of a quadratic function is graph of a quadratic function is symmetric.symmetric.

x = hx = h

where h is the x-coordinate of the where h is the x-coordinate of the vertex.vertex.



So, if we want to examine the So, if we want to examine the characteristics of the graph of a characteristics of the graph of a quadratic function, our job is to quadratic function, our job is to transform the standard formtransform the standard form

f(x) = axf(x) = ax22 + bx + c + bx + c

into the forminto the form

f(x) = a(x – h)f(x) = a(x – h)22 + k + k



This will require to process of This will require to process of completing the square.completing the square.

GRAPHING QUADRATIC FUNCTIONS

GRAPHING QUADRATIC FUNCTIONS

Graph the functions below by Graph the functions below by hand by determining whether its hand by determining whether its graph opens up or down and by graph opens up or down and by finding its vertex, axis of finding its vertex, axis of symmetry, y-intercept, and x-symmetry, y-intercept, and x-intercepts, if any. Verify your intercepts, if any. Verify your results using a graphing results using a graphing calculator.calculator.

f(x) = 2xf(x) = 2x22 - 3 - 3 g(x) = xg(x) = x22 - 6x - 6x - 1- 1

h(x) = 3xh(x) = 3x22 + 6x + 6x k(x) = -2xk(x) = -2x22 + + 6x + 26x + 2

DERIVING THE FORMULA FOR THE

VERTEX

DERIVING THE FORMULA FOR THE

VERTEX

A formula for the x-coordinate of A formula for the x-coordinate of the vertex can be found by the vertex can be found by completing the square on the completing the square on the standard form of a quadratic standard form of a quadratic function.function.

f(x) = axf(x) = ax22 + bx + c + bx + c

CHARACTERISTICS OF THE GRAPH OF A QUADRATIC

FUNCTION

CHARACTERISTICS OF THE GRAPH OF A QUADRATIC

FUNCTION

f(x) = axf(x) = ax22 + bx + c + bx + c

2ab-

x :SYMMETRY OF AXIS 2ab-

f, 2ab-

VERTEX

Parabola opens up if a > 0.Parabola opens up if a > 0.

Parabola opens down if a < 0.Parabola opens down if a < 0.

EXAMPLEEXAMPLE

Determine without graphing Determine without graphing whether the given quadratic whether the given quadratic function has a maximum or function has a maximum or minimum value and then find the minimum value and then find the value. Verify by graphing.value. Verify by graphing.

f(x) = 4xf(x) = 4x22 - 8x + 3 - 8x + 3 g(x) = -2xg(x) = -2x22 + + 8x + 38x + 3

THE X-INTERCEPTS OF A QUADRATIC

FUNCTION

THE X-INTERCEPTS OF A QUADRATIC

FUNCTION1.1.If the discriminant bIf the discriminant b22 – 4ac > 0, the – 4ac > 0, the

graph of f(x) = axgraph of f(x) = ax22 + bx + c has two + bx + c has two distinct x-intercepts and will cross distinct x-intercepts and will cross the x-axis twice.the x-axis twice.

2. If the discriminant b2. If the discriminant b22 – 4ac = 0, the – 4ac = 0, the graph of f(x) = axgraph of f(x) = ax22 + bx + c has one + bx + c has one x-intercept and touches the x-axis at x-intercept and touches the x-axis at its vertex.its vertex.

3. If the discriminant b3. If the discriminant b22 – 4ac < 0, the – 4ac < 0, the graph of f(x) = axgraph of f(x) = ax2 2 + bx + c has no x-+ bx + c has no x-intercept and will not cross or touch intercept and will not cross or touch the x-axis.the x-axis.

FINDING A QUADRATIC FUNCTION

FINDING A QUADRATIC FUNCTION

Determine the quadratic function Determine the quadratic function whose vertex is (1,- 5) and whose y-whose vertex is (1,- 5) and whose y-

intercept is -3.intercept is -3.

CONCLUSION OF SECTION 2.4CONCLUSION OF SECTION 2.4

section 2.4 the graph of a quadratic function the graph of a quadratic function

Documents

quadratic function fx

quadratic function slide

graph of fx

graphing quadratic functions

c slide

given quadratic function

simplest quadratic function

right slide