g raphing a q uadratic f unction a quadratic function has the form y = ax 2 + bx + c where a 0

GRAPHING A QUADRATIC FUNCTION

A quadratic function has the form

y = ax 2 + bx + c where a 0.


The graph is “U-shaped” and is called a parabola.


The highest or lowest point on the parabola is called the ver tex.


In general, the axis of symmetry for

the parabola is the vertical line

through the vertex.


These are the graphs of y = x 2

and y = x 2.


The origin is the vertex for both graphs.

The origin is the lowest point on the

graph of y = x 2, and the highest point

on the graph of y = x 2.


The y-axis is the axis of symmetryfor both graphs.

THE GRAPH OF A QUADRATIC FUNCTION


CONCEPT

SUMMARY

• The axis of symmetry is the vertical line x = – .b2a

The graph of y = a x 2 + b x + c is a parabola with these

characteristics:

• The parabola opens up if a > 0 and opens down if a < 0.

The parabola is wider than the graph of y = x 2 if a < 1 and

narrower than the graph y = x 2 if a > 1.

• The x-coordinate of the vertex is – .b2a

Graph y = 2 x 2 – 8 x + 6

SOLUTION

Note that the coefficients for this function are a = 2, b = – 8, and c = 6.

Since a > 0, the parabola opens up.

Graphing a Quadratic Function


Graph y = 2 x 2 – 8 x + 6

x = – = – = 2 b2 a

– 82(2)

y = 2(2)2 – 8 (2) + 6 = – 2

So, the vertex is (2, – 2).

(2, – 2)

The x-coordinate is:

The y-coordinate is:

Find and plot the vertex.

(2, – 2)


Graph y = 2 x 2 – 8 x + 6

Draw a parabola through the plotted points.

(0, 6)

(1, 0)

(4, 6)

(3, 0)

Draw the axis of symmetry x = 2.

Plot two points on one side of theaxis of symmetry, such as (1, 0)and (0, 6).

Use symmetry to plot two more points, such as (3, 0) and (4, 6).

VERTEX AND INTERCEPT FORMS OF A QUADRATIC FUNCTION


FORM OF QUADRATIC FUNCTION CHARACTERISTICS OF GRAPH

Vertex form:

Intercept form:

y = a (x – h)2 + k

y = a (x – p )(x – q )

For both forms, the graph opens up if a > 0 and opens down if a < 0.

The vertex is (h, k ).

The axis of symmetry is x = h.

The x -intercepts are p and q.

The axis of symmetry is half-way between ( p , 0 ) and (q , 0 ).

Graphing a Quadratic FunctionGraphing a Quadratic Function

Graph y = – (x + 3)2 + 412

To graph the function, first plot the vertex (h, k) = (– 3, 4).

(– 3, 4)SOLUTION The function is in vertex form

y = a (x – h)2 + k.

a = – , h = – 3, and k = 4

12

a < 0, the parabola opens down.

Graphing a Quadratic FunctionGraphing a Quadratic Function in Vertex Form

Use symmetry to completethe graph.

(– 3, 4)

(1, – 4)

(–1, 2)

(– 7, – 4)

(– 5, 2)

Graph y = – (x + 3)2 + 412

Draw the axis of symmetryx = – 3.

Plot two points on one side of it, such as (–1, 2) and (1, – 4).

Graphing a Quadratic Function in Intercept Form

Graph y = – ( x +2)(x – 4)

The quadratic function is in intercept form y = a (x – p)(x – q), where a = –1, p = – 2, and q = 4.

SOLUTION


Graph y = – ( x +2)(x – 4)

The axis of symmetry lies half-way between these points, at x = 1.

(– 2, 0) (4, 0)

The x-intercepts occur at (– 2, 0) and (4, 0).(– 2, 0) (4, 0)


Graph y = – ( x +2)(x – 4)

So, the x-coordinate of the vertex is x = 1 and the y-coordinate of the vertex is:

y = – (1 + 2) (1 – 4) = 9 (– 2, 0) (4, 0)

(1, 9)


You can change quadratic functions from intercept form or vertex form to standard form by multiplying algebraicexpressions.

One method for multiplying expressions containing two terms is FOIL. Using this method, you add the products of the First terms, the O uter terms, the Inner terms, and the Last terms.


First

FF OO II LL

= x 2 + 8 x + 15+ + +x

2 5x 3x 15( x + 3 )( x + 5 ) =

OuterInnerLast

F O I L

g raphing a q uadratic f unction a quadratic function has the form y = ax 2 + bx + c where a 0

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