4.1 Graph Quadratic Functions in Standard Form

 

Quadratic function

A function that can be written in the standard form y = ax2 + bx+ c where a ¹ 0

 

Parabola

The U-shaped graph of a quadratic function

 

Vertex

The lowest or highest point on a parabola

 

Axis of symmetry

The vertical line that divides the parabola into mirror images and passes through the vertex

 

Minimum and maximum value

For y= ax2 + bx+ c, the vertex's y-coordinate is the minimum value of the function if
a > 0 and its maximum value if a < 0.

 

PARENT FUNCTION FOR QUADRATIC FUNCTIONS

The lowest or highest point on a parabola is the vertex for
f(x) = x2 is (0,0)
 

The axis of symmetry divides the parabola into mirror images and passes through the vertex
 
The parent function for the family of all quadratic functions is f(x) = __x2__ . The graph is shown below

For f(x) = ax2, and for any quadratic function g(x)= ax2 + bx + c where b = 0, the vertex lies on the __y-axis__ and the axis of symmetry is x = _0_.

 


 

Example 1

Graph a function of the form y = ax2 + c

 

Graph y = -2x2 + 2. Compare the graph with the graph of y = x2.

 

Solution

1.     Make a table of values for y = -2x2 + 2.

 

x

-2

-1

0

1

2

y

_-6_

_0_

_2_

_0_

_-6_

Choose values of x on both sides of the axis of symmetry
x = 0.
 

 

2.     Plot the points from the table.

3.     Draw a smooth __curve__ through the points.

 

 


4.     Compare the graphs of y = -2x2 + 2 and y = x2. Both graphs have the same __axis of symmetry__. However, the graph of y = -2x2 + 2 opens _down_ and is _narrower_ than the graph of y = x2. Also, its vertex is _2_units higher.

 

PROPERTIES OF THE GRAPH OF y = ax2 + bx + c

Characteristics of the graph of y = ax2 + bx + c:

·       The graph opens up if a _>_ 0 and opens down if a _<_ 0.

·       The graph is narrower than the graph of y = x2 if | a | _>_ 1 and wider if | a | _<_ 1.

·       The axis of symmetry is x =               and the vertex has x-coordinate            .

·       The y-intercept is _c_. So, the point (0, _c_) is on the parabola.

 

Example 2

Graph a function of the form y = ax2 + bx + c

 

Graph y = -x2 + 4x - 3.

 

Solution

 

1.    Identify the coefficients of the function. The coefficients are a = _-1_, b = _4_, and
c = _-3_. Because a _<_ 0, the parabola opens _down_.

2.    Find the vertex. First, calculate the x-coordinate.

Be sure to include the negative sign before the fraction when calculating the x-coordinate of the vertex.
 
 

 


x =            =                = _2_

Then find the y-coordinate.

y = __-(2)2 + 4(2) - 3__ = _1_

The vertex is (_2_, _1_). Plot this point.

3.    Draw the axis of symmetry x = _2_.

4.    Identify the y-intercept c, which is _-3_.

Plot the point (0, _-3_). Then reflect this point in the axis of symmetry to plot another point (4, _-3_).

5.    Evaluate the function for another value of x, such as x = 1.

y = __-(1)2 + 4(1) - 3__ = _0_

Plot the point (1, _0_) and its reflection (3, _0_) in the axis of symmetry.

6.     Draw a parabola through the plotted points.

MINIMUM AND MAXIMUM VALUES

Words For y = ax2 + bx + c, the vertex's y-coordinate is the minimum value of the function if a _>_ 0 and the maximum value if a _<_ 0.

Example 3

Find the minimum or maximum value

 

Tell whether the function y = -3x2 + 12x - 6 has a minimum value or a maximum value. Then find the minimum or maximum value.

 

Solution

Because a _<_ 0, the function has a __maximum__ value. To find it, calculate the coordinates of the vertex.

x =            =                     =     _2_

y = __-3(2)2 + 12(2) - 6__ = _6_

The maximum value is y = _6_ .