1. Lesson 2

1.7. Explore 3

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Sketching Quadratic Functions Using Transformations

 

In Try This 3 you saw how the graph of was drawn by transforming the graph of y = x2. You used your understanding of the variables a, p, and q to make the transformations.

 

When you found the coordinates of the vertex, did you see the rule you identified in Try This 1? The rule you developed in Try This 1 showed that the vertex could be found from the values of p and q in y = a(x − p)2 + q where,

 

  • p is the x-coordinate of the vertex

  • q is the y-coordinate of the vertex

  • the vertex coordinate = (p, q)


In Try This 2 you developed a rule to identify the number of x-intercepts. You may have noticed the following:

 

  • If q = 0, only 1 x-intercept is present.

  • If q ≠ 0,

    • a and q are the same sign and there are no x-intercepts

    • a and q are different signs and there are two x-intercepts

 

This is a play button that opens Variables a and q and Number of x-Intercepts.

Watch Variables a and q and Number of x-Intercepts now.



caution

Be careful when you are finding the value of p and q from y = a(x − p)2 + q.

 

 

Remember that the sign before p in the equation is negative, and the sign before q is positive.



Self-Check 2
  1. Sketch the graph of y = 4(x − 1)2 using transformations. Use the graph paper provided or a graphing tool. Answer

  2. Use the quadratic function y = −0.4(x − 3)2 − 1 to answer the following:

    1. Identify the

      1. vertex
      2. direction of opening
      3. axis of symmetry
      4. domain and range
      5. number of x-intercepts and what the x- and y-intercepts are, if any

      Answer

    2. Sketch the graph of the function using transformations. Use the graph paper provided or a graphing tool.

      Answer

Did You Know?


Although you have written the vertex form of the quadratic function as y = a(x − p)2 + q, the quadratic function can also properly be written in function notation as f(x) = a(x − p)2 + q. If this form is used, the vertical axis on the graph would need to be labelled as f(x), as shown in the graph.

 

This shows the coordinate plane with x-values from −7 to 8 and y-values from −7 to 8. The graph of f of x equals negative 0.4 quantity x minus 3 squared plus 4 is overlaid.