Quadratic equations in vertex form

  1. Using the sliders, set h = 0 and k = 0. (To quickly set a slider to a specific number, type the number into the field to the right of the slider, and then press ENTER.) Vary the value of a and observe how the graph changes.
    1. In which direction does the parabola open when a is positive? In which direction does it open when a is negative?
    2. What does the graph look like when a = 0? Use the equation y = a(x − h)2 + k to explain why this occurs.
    3. As the absolute value of a increases, does the graph become more or less steep? Does this make the parabola appear narrower or wider? Explain why increasing the absolute value of a has this effect on the graph.
  2. Set k = 0 and a = 1. Vary the value of h and observe how the graph changes.
    1. How does the graph change as the value of h changes?
    2. What effect does the value of h have on the shape of the graph?
  3. Set h = 0 and a = 1. Vary the value of k and observe how the graph changes.
    1. How does the graph change as k changes?
    2. What effect does the value of k have on the shape of the graph?
  4. Turn on Show vertex and intercept(s). Depending on whether the parabola opens up or down, the vertex is either the highest or the lowest point on the graph.
    1. Which coordinate of the vertex does h correspond to, the x-coordinate or the y-coordinate?
    2. Which way does the graph move when h increases? When h decreases?
    3. Which coordinate of the vertex does k correspond to, the x-coordinate or the y-coordinate?
    4. Which way does the graph move when k increases? When k decreases?
  5. For each of the following, write an equation that matches the parabola described. Then check your answers by graphing each equation in the Gizmotm.
    1. Write an equation for a parabola that opens up and has a vertex at (2, 0).
    2. Write an equation for a parabola that opens up and has a vertex at (0, −3).
    3. Write an equation for a parabola that opens down and has a vertex at (−1, 2).

The vertex and intercepts of a parabola

When graphing a parabola, it is useful to know the location of its vertex and intercepts. A parabola that opens down has a maximum y-value at its vertex. A parabola that opens up has a minimum y-value at its vertex.

  1. Using the Gizmo, find three different quadratic equations whose parabolas have a maximum y-value of 3.
    1. What must be true about the value of a in order for the maximum y-value to be 3?
    2. What must be true about the value of k in order for the maximum y-value to be 3?
  2. Using the Gizmo, find three different quadratic equations whose parabolas have a minimum y-value of 2.
    1. What must be true about the value of a in order for the minimum y-value to be 2?
    2. What must be true about the value of k in order for the minimum y-value to be 2?
  3. A parabola can have 0, 1, or 2 x-intercepts. Vary the values of a and k to graph each of the following parabolas.
    1. Graph a parabola that opens up and has no x-intercepts. In this case, are the values of a and k positive or negative? Repeat this process for a parabola that opens down?
    2. Graph a parabola that opens up and has exactly one x-intercept. In this case, what do you notice about the value of k? Repeat this process for a parabola that opens down.
    3. Graph a parabola that opens up and has two x-intercepts. In this case, are the values of a and k positive or negative? Repeat this process for a parabola that opens down.
  4. A table is another way to identify critical points on a parabola. Use the sliders to graph y = (x − 1)2 − 4. Click on the TABLE tab to examine a set of points on the curve.
    1. For any function, what will the x-value be at the y-intercept?
    2. Using this fact, how can you find the y-intercept(s) of the function y = (x − 1)2 − 4 in the table?
    3. Using the same approach, how would you find the x-intercept(s) of the function y = (x − 1)2 − 4 in the table?
    4. How can you find the vertex of a parabola using a table? What is the vertex of y = (x − 1)2 − 4?