1. Lesson 4

1.11. Lesson 4 Summary

Mathematics 20-2 M3 Lesson 4

Module 3: Quadratics

 
Lesson 4 Summary
 

In this lesson you studied a specific form of the quadratic function y = a(x − h)2 + k called the vertex form. You discovered that a quadratic function written in vertex form has the following characteristics.

  • The vertex of the parabola has the coordinates (h, k). The value of h gives the position of the vertex of the parabola relative to the x-axis. The value of k gives the position of the vertex of the parabola relative to the y-axis. (That is one reason this form of the function is called the vertex form.)

  • The equation of the axis of symmetry of the parabola is x = h.

  • The value of a influences the sharpness of the curve of the parabola. The parabola becomes narrower as a becomes more positive or negative.

  • If the value of a is negative, the parabola will open downwards and the function has a maximum value of k when x = h.

  • If the value of a is positive, the parabola will open upwards and the function has a minimum value of k when x = h.
This shows a graph under the heading y = a(x – h) quantity squared plus k and a > 0. The parabola opens upwards and has a vertex labelled (h, k) and an axis of symmetry of x = h. This shows a graph under the heading y = a(x � h) quantity squared plus k and a < 0. The parabola opens downwards and has a vertex labelled (h, k) and an axis of symmetry of x = h.

from: CANAVAN-MCGRATH ET AL. Principles of Mathematics 11, © 2012 Nelson Education Limited. p. 323. Reproduced by permission.

 

You learned to apply these ideas to find answers to problems involving the paths of snowmobile jumps. The vertex told how high the jump was and where that maximum occurred. The axis of symmetry was half the trajectory, so doubling it gave the maximum distance. Substituting values in for x gave the height at various distances.

 

This shows a photo of a snowmobile jumped into the air.

Hemera/Thinkstock

 

You also confirmed that the vertex form of the quadratic function can be used to determine the number of x-intercepts.

 

This shows three graphs. In all three graphs, the parabola opens upwards. The first graph has two x-intercepts. The vertex is below the x-axis. It reads “a and k have opposite signs.” The second graph has one x-intercept. The vertex is on the x-axis. It reads “k = zero.” The third graph has no x-intercepts. The vertex is above the x-axis. It reads “a and k have the same signs.”

CREDIT: From: CANAVAN-MCGRATH ET AL. Principles of Mathematics 11, © 2012 Nelson Education Limited. p. 362. Reproduced by permission.

 

In the next lesson you will investigate another form of the quadratic function called the factored form.