GRAPHING PARABOLASPAGE 1

The graph of an equation in the form y = Ax2 + Bx + C, where A ≠ 0, is called a parabola. In this activity you will investigate how the values of Aand Caffects the shape and position of the parabola.

  1. Let us begin by considering the special case y = Ax2 where B = 0 and C = 0. Use your graphing calculator to graph each equation. Carefully sketch each graph displayed on the axes provided.

a. y = 0.25x2 / b. y = 0.5x2
c. y = 2x2 / d. y = 6x2
e. y = -0.25x2 / f. y = -0.5x2
g. y = -2x2 / h. y = -6x2
  1. Use your sketches from page 1 to help you answer the following questions:
  2. What common shape do all of the graphs share? ______
  3. Under what condition does the graph of y = Ax2 open upward?

______Open downward? ______

  1. As A increases in absolute value (distance from zero), what happens to the graph of y = Ax2?

______

  1. Next consider the equation y = Ax2 + Bx + C, where B = 0. In this case the equation becomes y = Ax2 + C. Use your graphing calculator to produce the graphs of the following equations. Carefully sketch the graphs from the calculator display onto the corresponding axes below making sure to include the vertex (top/bottom of the graph).

a. y = 2x2 + 3 / b. y = 2x2 – 3
c. y = 2x2 – 4 / d. y = 2x2 + 1
  1. Use your sketches from above to answer the following questions:
  2. How does the value of C affect the graph of y = 2x2 + C?

______

  1. The graph of y = 0.5x2 – 3 intersects the y-axis at the point (___ , ___).
  2. The y-intercept of the graph y = 2x2 + 4 is the point (___ , ___). Check your answers with your graphing calculator.
  1. Without using your graphing calculator, indicate whether the graph of each of the following equations will open upward or downward; decide if it will be relatively narrow or wide; determine the y-intercept; and sketch a graph of each.

Equation / Up/Down / Wide/Narrow / y-intercept / Graph
y = -0.6x2 – 4 /
y = 5x2 + 3 /
y = -2x2 + 2 /
y = 0.3x2 - 3 /

Now use your graphing calculator to DOUBLE CHECK your answers and sketches!

  1. For each of the following equations, use your graphing calculator to graph the equation; carefully sketch the parabola on the corresponding axes, noting the y-intercepts and also the x-intercepts; and use the space provided to calculate the discriminant which can be calculated using the formula B2 – 4AC and record the value in the space provided.

Equation / Graph / Discriminant (B2 – 4AC) / # x-intercepts
y = x2 + 4x + 4 /
y = -8x2+ 1 /
y = x2 – 2x + 3 /
y = -x2 + 2x – 1 /
y = 2x2 - 7x + 3 /
y = -2x2 + x - 1 /
  1. Use your results from exercise 6 on the previous page to answer the following questions.
  2. When will the graphs of y = Ax2 +Bx + C have exactly one x-intercept? ______
  3. When will the graphs of y = Ax2 +Bx + C have two x-intercepts? ______
  4. When will the graphs of y = Ax2 +Bx + C have no x-intercepts? ______
  5. Complete each row of the chart without using your graphing calculator. Before continuing to the next for graph the equation using your graphing calculator and check if the characteristics of the displayed graph agree with your predictions.

Equation / Up/Down / Wide/Narrow / y-intercept / Discriminant / # x-intercepts
y = 0.5x2 - 2x – 6
y = 6x2 +2 x + 1
y = -x2 + 4x – 4
y = -3x2 + x – 2
y = -4x2 +4 x + 1
y = 2x2 - 8x + 8

Summary:

We just looked at the properties of A and C in the quadratic equation y = Ax2 + Bx + C. Answer the following questions below.

  1. Explain what happens to the graph of a quadratic when the value of A changes.

______

2. Explain what happens to the graph of a quadratic when the value of C changes.

______

  1. Looking back at all of your graphs what happens to the graph of a quadratic when the value of B changes?

______