1. Lesson 1

1.7. Explore 2

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Maximum and Minimum Points

 

In Axis of Symmetry you may have noticed the pattern between a and the existence of a maximum or minimum point. The following chart summarizes the pattern.

 

 
a Sketch Orientation Maximum or Minimum
3 This illustration shows the graph of y = 3x squared, which is a parabola with a vertex at the origin and that opens upward. opens upward minimum
−3 This illustration shows the graph of y = -3x squared, which is a parabola with a vertex at the origin and that opens downward. opens downward maximum

 

You may have found that the value of a in y = a(x − p)2 + q has an effect on the parabola. The value of a in the equation influences the sharpness of the curve of the parabola. The table shown summarizes this effect.

 

 
Value of a Effect on Parabola

 

As the value of a increases above 0 . . .
This illustration shows an upward-opening parabola with arrows from the sides of the parabola going inward.

 

. . . the curve goes from very wide to more and more narrow.

 

As the value of a becomes increasingly negative . . .
This illustration shows a downward-opening parabola with arrows from the sides of the parabola going inward.

 

. . . the downward-opening curve becomes more and more narrow.

 

You also explored the value of p and q in the vertex form of a quadratic function.

 

 
q
  • The value of q in the equation gives the position of the vertex of the parabola relative to the y-axis (the q-value moves the vertex up and down).

  • That is one reason this form of the equation is called the vertex form.
 This illustration shows a red vertical line on a set of coordinate axes.
p
  • The value of p in the equation gives the position of the vertex of the parabola relative to the x-axis (the p-value moves the vertex left and right).

  • The value of p is also the x-coordinate of the axis of symmetry of the parabola.
 This illustration shows a horizontal red line on a set of coordinate axes.

 

Did You Know?

The shape of the reflector inside flashlights and automobile headlamps is a parabola. This shape formed by a quadratic function reflects the light out in a concentrated forward beam rather than in a scattered pattern.

 

This is a photo of a car headlight.

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