Lesson 4

Explore

How can the vertex form of the quadratic function help you to answer questions about the trajectory of snowmobiles that are being used in a competition for the longest jump? The vertex form easily identifies the coordinates of the vertex, the direction of opening, the maximum and minimum positions, and the equation of the axis of symmetry.

In the Discover section you learned that if the equation is in vertex form, y = a(x − h)2 + k, the vertex is at (h, k) and this is the minimum value (if the graph opens up) or maximum value (if the graph opens down) for the function.

When h is a positive value, the graph of y = x2 moves to the right h units; and when h is negative, the graph moves to the left h units. When k is a positive value, the graph of y = x2 moves up k units; and when k is negative, the graph moves down k units.

You have discovered that a determines whether the graph opens upwards or downwards and how wide it will open. The equation of the axis of symmetry of the parabola is x = h.

from: CANAVAN-MCGRATH ET AL. Principles of Mathematics 11, © 2012 Nelson Education Limited. p. 326. Reproduced by permission.

Because the vertex form gives you the direction of opening and the vertex of the parabola, the graph of the quadratic function can be sketched more easily using the vertex form than the standard form.

Watch the animation Chris’s Jump to see how the vertex form of the quadratic function can help answer questions about the trajectory of Chris’s jump on a snowmobile.

Ryan McVay/Lifesize/Thinkstock

Self-Check 1

For one of Jean-Guy’s jumps, the quadratic function in vertex form describing the trajectory is y = −0.2(x − 3)2 + 1.8. As with Chris’s jump, x is the distance from the take-off point, in metres, and y is the height in metres above the level of the meadow.

1. What would a sketch of the graph of Jean-Guy’s trajectory look like? Answer
   
   
2. What was the height of Jean-Guy’s jump? Answer
   
   
3. What was the distance of Jean-Guy’s jump? Answer
   
   
4. What domain and range of the function is reasonable? Answer
   
   
5. Did Chris or Jean-Guy have the larger jump? Answer
   
   
6. What was Jean-Guy’s height when he was 4 m from the take-off point? Answer

Use the Quadratic Function applet to explore conversions from vertex form to general form and vice versa.

Move the red v(x) sliders to graph v(x) = x2 in vertex form y = (x + 0)2 + 0. Continue by moving the blue f(x) sliders to graph f(x) = x2 + 0x + 0. Do the red and blue graphs overlap? Can you graph the function f(x) = x2 − 4x + 7 using the blue sliders? Give it a try. Note the position of the vertex and the y-intercept. What function in the vertex form would match this graph? Go to the red vertex form sliders, and duplicate the blue graph. What values did you use for a, h, and k?

Change the vertex form to v(x) = 0.6(x + 2)2 + 1.6, and examine the graph and vertex carefully. Duplicate this graph using the blue f(x) sliders.

Choose different values for a, b, and c using the blue f(x) sliders, and then duplicate the graph using the red v(x) sliders.

Then choose new values for a, h, and k using the red v(x) sliders, and duplicate the graph using the f(x) blue sliders.

The vertex form of a quadratic function can be converted to the standard form by squaring the binomial, clearing the brackets, and collecting like terms. For Chris’s jump, view the process in Chris’s Equation.