Quadratic Functions in Vertex Form
This is a quadratic function expressed in vertex form: f(x) = a(x - h)² + k.
The graph of a quadratic function produces a parabola. Every parabola has either a maximum or minimum value located at its vertex.
Parabolas are symmetrical. The line that divides the parabola into two equal halves is called the line of symmetry.
The location where a parabola touches the x-axis is called an x-intercept or root.
1. Click on the "Show equation" and "y-intercept" check boxes. Set "a" equal to 1, "h" equal to 1 and "k" equal to 0. How does the y-intercept compare to "h"? What happens when you set "h" to 2? Vary "h" and note any changes to the y-intercept. How does the value of "h" relate to the y-intercept?
2. Now, set "h" equal to one and adjust the value of "a." How does the value of "a" relate to the y-intercept?
3. Lastly, set "a" and "h" to 1 and vary "k". Considering the relation already established, how to you suppose k is related to the y-intercept. Set "a" to 2. Does your conjecture still hold? Adjust the values of a, h, and k and attempt to find an equation that relates all three values to the y-value of the y-intercept.
4. Adjust the value of "h" so that it is positive. Note which side of the y-axis the vertex is on. Use the slider bar to make "h" negative. Again, note which side of the y-axis the vertex is on. Lastly, set "h" to zero and note the location of the vertex. What do you conclude?
5. Reset the construction by pressing the reset button at the top. Click on "Vertex." Note the values of "h" and "k" and the coordinates of the vertex. Use the slider bars to adjust the values of "h" and "k" and note any changes to the vertex. What do you conclude?
6. Click on "h" and "Line of symmetry." Determine the relation between the two. Explain your findings.
7. Reset the construction by pressing the reset button at the top and click on "Roots." Adjust the value of "k" noting the relationship between the two. How many roots are there when k is negative? Zero? Positive? What do you surmise?
8. Reset the construction again by pressing the reset button at the top. Note the shape of the graph. This equation y = x² is the parent function for quadratic functions. Increase the value of "a" and note the difference between the new function and the parent function. Does increasing the value of "a" stretch (make wider) or shrink (make narrower) the graph of the function?
9. Reset "a" to one so you can note the parent function once again. Adjust "a" so that it is between zero and one. What can you conclude about the effect on the graph for values of "a" in the range from zero to one compared to the parent function?
10. Finally, use the slider bar to make "a" a negative number. What does changing the value of "a" to a negative number do to the graph?