Unit 4, Lesson 12
Graphing the Standard Form (Part 1)
Integrated Math 2
12.1
Warm-up
Which graph corresponds to which equation? Explain your reasoning.
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Which graph corresponds to which equation? Explain your reasoning.
Remember that the graph representing any quadratic function is a shape called a parabola. People often say that a parabola “opens upward” when the lowest point on the graph is the vertex (where the graph changes direction), and “opens downward” when the highest point on the graph is the vertex. Each coefficient in a quadratic expression written in standard form tells us something important about the graph that represents it.
The graph of is a parabola opening upward with vertex at . Adding a constant term 5 gives and raises the graph by 5 units. Subtracting 4 from gives and moves the graph 4 units down.
| -3 | -2 | -1 | 0 | 1 | 2 | 3 | |
| 9 | 4 | 1 | 0 | 1 | 4 | 9 | |
| 14 | 9 | 6 | 5 | 6 | 9 | 14 | |
| 5 | 0 | -3 | -4 | -3 | 0 | 5 |
A table of values can help us see that adding 5 to increases all the output values of by 5, which explains why the graph moves up 5 units. Subtracting 4 from decreases all the output values of by 4, which explains why the graph shifts down by 4 units.
In general, the constant term of a quadratic expression in standard form influences the vertical position of the graph. An expression with no constant term (such as or ) means that the constant term is 0, so the -intercept of the graph is on the -axis. It’s not shifted up or down relative to the -axis.
The coefficient of the squared term in a quadratic function also tells us something about its graph. The coefficient of the squared term in is 1. Its graph is a parabola that opens upward.
Coordinate plane, 4 graphs of quadratic functions, all with the maximum or minimum at the origin. First, y = 2 x squared, opens up. Second, y = x squared, opens up but wider than the first. Third, y = fraction 1 over 2 x squared, opens up wider than the first 2. Fourth, y = negative 2 x squared, opens down.
| -3 | -2 | -1 | 0 | 1 | 2 | 3 | |
| 9 | 4 | 1 | 0 | 1 | 4 | 9 | |
| 18 | 8 | 2 | 0 | 2 | 8 | 18 | |
| -18 | -8 | -2 | 0 | -2 | -8 | -18 |
If we compare the output values of and , we see that they are opposites, which suggests that one graph would be a reflection of the other across the -axis.