Lesson 4

Launch

Before beginning this lesson, you should be able to

• write the trinomial form of a binomial squared
  
  
• factor a trinomial that is a perfect square

Are You Ready?

Write the expanded form of each of the following quadratic functions by squaring the binomial to form a trinomial:

1. y = (x + 4)2 Answer
   
   
2. f(x) = (x − 5)2 Answer
   
   
3. y = 4(x + 3)2 Answer

If you successfully completed the questions, move on to the Discover section of the lesson.

If you experienced difficulties, use the resources in the Refresher section to review the concepts before continuing through the lesson.

Refresher

View the video “Square a Binomial.”

Khan Academy CC BY-NC-SA 3.0

Go back to the Are You Ready? section, and try the questions again. If you are still having difficulty, contact your teacher.

Discover

In Lesson 1 you investigated how changing the coefficients and constant in the standard form of a quadratic function affects the shape of the graph. From these observations, you were able to identify characteristics of the graphs of quadratic functions written in standard form, y = ax2 + bx + c.

You discovered the following characteristics of a graph of a quadratic function that is defined by the equation y = ax2 + bx + c.

• When a is greater than zero, the parabola opens up and the vertex is a minimum.
  
  
• When a is less than zero, the parabola opens down and the vertex is a maximum.
  
  
• The constant term, c, is the value of the parabola’s y-intercept.
  
  
• The parabola is symmetrical about the vertical axis of symmetry.
  
  
• The vertex is a point on the axis of symmetry.

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

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

So, how does changing the values of a, h, and k in a quadratic function written in vertex form, y = a(x − h)2 + k, affect the shape and position of the parabola? The answer to this question can help determine the characteristics of a quadratic function written in vertex form.

Try This 1

Use the applet Quadratic Function: Vertex Form to investigate how changing the values of a, h, and k for a quadratic function written in vertex form, y = a(x − h)2 + k, affects the shape and the position of the parabola.

Complete the following questions.

1. Set the values of h and k to zero. Set the a slider to 1. You should have the graph of the function
   y = x2. Use the a slider to change the coefficient of x2. Try both positive and negative values
   for a.
   
   
   1. How do the parabolas change as you change the value of a?
      
      
   2. For each function you graphed in part a, determine the coordinates of the vertex and the equation of the axis of symmetry.
      
      
2. Set the slider for a to +1, and set the value of k to 0. Use the h slider to investigate how changing the value of h in the quadratic function y = a(x − h)2 + k affects the position of the graph. Try both positive and negative values for h.
   
   
   1. How do the parabolas change as you change the value of h?
      
      
   2. How do the coordinates of the vertex and the equation of the axis of symmetry change as you change the value of h?
      
      
3. Set the slider for a to +1, and set the value of h to 0. Use the k slider to see how changing the value of k in the quadratic function y = a(x − h)2 + k affects the position of the graph. Try both positive and negative values for k.
   
   
   1. How do the parabolas change as you change the value of k?
      
      
   2. How do the coordinates of the vertex and the equation of the axis of symmetry change as you change the value of k?
      
      
4. Make a conjecture about how the values of a, h, and k determine the characteristics of a parabola.
   
   
5. Test your conjecture. Drag the sliders for a, h, and k to create the quadratic functions shown in the Vertex Table. For each function, determine the coordinates of the vertex and the equation of the axis of symmetry.

Save your completed Vertex Table to your course folder.

Share 1

Based on your observations from Try This 1, discuss the following questions with another student or appropriate partner.

1. How does the position of the graph change as the value of h changes? Describe the pattern you see.
   
   
2. How does the position of the graph change as the value of k changes? Describe the pattern you see.
   
   
3. Why do you suppose this form of the quadratic function is called the vertex form?

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.

Self-Check 2

Convert the vertex form of Jean-Guy’s jump, y = −0.2(x − 3)2 + 1.8, to the standard form by showing each step. Answer

You are given the vertex of a parabola and the intercepts and asked to write the function in vertex form. The h and k values are easily written from the vertex, but how do you get the value for a?

Try This 2

For the following graph, the vertex is at (−2.5, −4.9). From this information, the function can be written as y = a(x + 2.5)2 − 4.9.

1. Substitute the x- and y-values from one of the intercepts into the function and solve for a. What value for a do you get?
   
   
2. Verify by substituting the x-value from the second intercept and the a-value you just calculated into the function y = 0.4(x + 2.5)2 - 4.9. Did you get the y-value of the second intercept—in this case, y = 0?

Self-Check 3

1. For the following graph, write the function in vertex form. Answer
   
   
   
   
2. A children’s wading pool that has a parabolic profile is located at a playground. The depth of the pool, in metres measured from the edge of the pool, is modelled by the function y = 0.012(x – 5)2 – 0.3.
   
   
   1. What is the axis of symmetry of the graph?
      
      
   2. What is the maximum depth of the pool?
      
      
   3. How wide is the pool?
      
      
   4. If the water level was down by 0.05 m, how wide would the top of the water be? (Make sure to round your answer to two decimal places.)
      
      
   5. Officials want to paint a stripe around the pool at a depth of 0.15 m to give parents and toddlers a visual reminder of the increasing depth. How far from the edge of the pool should the line be painted?
      
      Answer

Try This 3

Use the gizmo titled “Quadratics in Vertex Form—Activity A” to see if you can predict whether a quadratic function will have zero, one, or two x-intercepts if the function is expressed in vertex form. Use the sliders to vary the values of a, h, and k. Choose “Show vertex and intercepts(s)” to see the coordinates of the vertex and the x- and y-intercepts for each function that you create.

Screenshot reprinted with permission of ExploreLearning.

Alternatively, you can manually create sketches of graphs of functions in vertex form using your graphing calculator or using the applet Quadratic Function: Vertex Form.

No matter which method you choose, be sure to organize the information you collect so you can look for connections or patterns. You may choose to organize your information in a chart, take screenshots of specific combinations of values, or sketch graphs of the quadratic functions in vertex form.

Self-Check 4

Connect

Lesson Assessment

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.

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.

Hemera/Thinkstock

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

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.