Investigation: Graphing Quadratic Equations

Part 1: The graph of y = x2

Step 1: Make a table and graph y = x2

X / Y
-2
-1
0
1
2

How much are the y’s going up by each time?

Step 2: Make a table and graph y = 2x2

X / Y
-2
-1
0
1
2

How much are the y’s going up by each time now? How does this compare to step 1 and why?

Step 3: Make a table and graph for y = (-1/2)x2

X / Y
-2
-1
0
1
2

How much are the y’s going up by now? How does this compare to step 1 and why?

So when graphing, once we know the vertex, to graph the rest of the parabola, we can use the pattern _____, ______, _____,… depending on what our a value is.

Part 2: Vertex Form:y = a(x – h)2 + k

Recall back to previous chapters. You already know how to graph equations such as

y = 2(x – 3)2 + 1y = -.5(x + 2)2– 4

So in y = a(x – h)2 + k, what does each part do?

a = ______

h = ______

k = ______

Graph each of the following. Does your graph have a minimum or a maximum? Estimate the zeroes using your graph. Then use your calculator to find the zeroes.

Part 3: Factored Form: y = a(x – r1)(x – r2)

Another easy way to graph quadratic equations are when they are factored.

Because we know roots (zeros, solutions) when the equation is set equal to zero, we can use these to plot points.

  • Set y = 0 and solve for x, using the zero-product property.
  • Those roots are the x-intercepts (r1, 0) and (r2, 0).
  • Find the axis of symmetry, half way between the roots.
  • Use that x-value to find the y-coordinate of the vertex.
  • Connect the points!
  • Check using 1, 3, 5..

Graph.

y = -2(x – 3)(x + 1)y = 1/3(x + 4)(x – 2)

Graph the following equation. Does your graph have a minimum or a maximum? What are the zeroes?

y = -1/2(x +3)(x - 1)

Part 4: Standard Form: y = ax2 + bx + c

The last way to graph is when an equation is in standard form.

  • Find the axis of symmetry by using .
  • Find the y-coordinate of the vertex.
  • Use 1, 3, 5..
  • Connect the points!

y = x2 – 2x – 3y = 3x2 + 12x + 8

Graph the following. Does your graph have a minimum or a maximum? Estimate the roots. Write in vertex form.

Using graphs to solve word problems

Example: Nora hits a softball straight up at a speed of 120 ft/s. Her bat contacts the ball at a height of 3 ft above the ground. An equation that can be used to represent the height of the ball as a function of time is y = -16x2 + 120x + 3.

  1. How high does the ball travel?
  1. When does the ball reach its maximum height?
  1. When does the ball hit the ground?
  1. When does the ball pass the point from which it was hit?

Example: A rock is thrown upward at an initial velocity of 100 m/s and an initial height of 25 m. An equation that relates the rock’s time and height is y = -4.9t2 + 100t + 25.

  1. When does the rock reach its maximum height?
  1. What is the maximum height?
  1. When does the rock hit the ground?
  1. When does the rock pass the point from which it was thrown?