Explore: Algebra Lab, Analyzing Linear Graphs
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See solution.
Let's begin by drawing a graph of a line that slants up and one of a line that slants down.
Now we can compare and contrast their key features!
| Key Feature | Slant Up | Slant Down |
|---|---|---|
| Domain | All real numbers | All real numbers |
| Range | All real numbers | All real numbers |
| Has x-intercept? | Yes | Yes |
| Has y-intercept? | Yes | Yes |
| Function value at the x-intercept | 0 | 0 |
| Has maximum or minimum? | No | No |
Now let's look at the differences. First we will study what happens as we move left on the graphs.
When we move to the left on a line slanting up we also move downwards. Moving to the left on a line that slants down will also make us move upwards. Next we will look at the graphs as we move right.
When we move to the right on a line slanting up, we at the same time move up. When we move to the right on a line slanting down we also move down. Another way to express this is that a line that slants up increases and that a line that slants down decreases across its entire domain. Let's study the values the function takes to the left and to the right of the x-intercept.
To the left of the x-intercept on a line that slants up it is negative, and a line that slants down is positive. To the right of the x-intercept it is the other way around. A line that slants up is there is positive, whereas a line that slants down is negative. Let's summarize the differences in a table.
| Key Feature | Slant Up | Slant Down |
|---|---|---|
| Direction as we move left | Down | Up |
| Direction as we move right | Up | Down |
| Increasing or decreasing | Increasing | Decreasing |
| Function values left of the x-intercept | Negative | Positive |
| Function values right of the x-intercept | Positive | Negative |