Activity
For this activity, you will use two graphs that represent situations you saw in other activities.
For questions 1 – 3, use the graph below: The graph represents , which describes the relationship between gallons of water in a tank and time in minutes.
Where on the graph can we see the 450?
Compare your answer:
450 is where the graph intersects the vertical axis.
Where can we see the -20?
Compare your answer:
−20 is the slope.
What do the numbers 450 and -20 mean in the situation that is graphed?
Compare your answer:
Your answer may vary, but here is a sample. 450 is the gallons of water in the tank at 0 minutes, before the tank starts to be drained. −20 means the gallons of water (vertical value) drops by 20 for every 1 minute increase in time (horizontal value).
The graph represents . It describes the relationship between pounds of almonds and figs and the dollar amount Clare spent on them. Suppose a classmate says, “I am not sure the graph represents because I don’t see the 6, 9, or 75 on the graph.” How would you show your classmate that the graph indeed represents this equation?
Compare your answer:
Your answer may vary, but here is a sample.
- If we substitute 0 for in the equation and solve for , we get or . That combination is the point or the -intercept of the graph. If we substitute 0 for in the equation and solve for , we get 12.5 for . That combination is the point (12.5,0) or the -intercept of the graph.
- If we substitute the - and -values of any point on the graph in the equation, the equation remains true.
- If we rewrite the equation and solve for , we have or . The matches where the graph intersects the -axis and matches the slope of the graph. (For every 3 additional pounds of almonds that Clare bought, she could buy 2 fewer pounds of figs.)
- From the graph, we can see that the -intercept is and the slope is , so the equation of the line is . Multiplying each side of the equation by 9 (an acceptable move) gives an equivalent equation, , which can be rewritten as .
Self Check
Additional Resources
Finding Intercepts from Graphs and Equations
The points where a line crosses the -axis and the -axis are called the intercepts of the line.
Let’s look at the graphs of the lines.
First, notice where each of these lines crosses the -axis.
Now, let’s look at the points where these lines cross the -axis.
Figure | The Line Crosses the -axis at: | Ordered Pair for this Point | The Line Crosses the -axis at: | Ordered Pair for this Point |
Figure (a) | 3 | 6 | ||
Figure (b) | 4 | -3 | ||
Figure (c) | 5 | -5 | ||
Figure (d) | 0 | 0 | ||
General Figure | ( | (0, |
-INTERCEPT and -INTERCEPT of a Line
The -intercept is the point where the line crosses the -axis.
The -intercept is the point where the line crosses the -axis.
Find the - and -intercepts from the Equation of a Line
Use the equation of the line. To find:
- the -intercept of the line, let and solve for .
- the -intercept of the line, let and solve for .
Example
Find the intercepts of
Solution
To find the -intercept:
Step 1 - Let .
Step 2 - Solve for .
Step 3 - Write the intercept as a point.
To find the -intercepts:
Step 1 - Let .
Step 2 - Solve for .
Step 3 - Write the intercept as a point.
Try it
Try It: Finding Intercepts from Graphs and Equations
Find the - and -intercept of .
Compare your answer: (8, 0) (0, 2)
Here is how to find the intercepts using a general strategy.
To find the -intercept,
Step 1 Let .
Step 2 - Solve for .
Step 3 - Write the intercept as a point.
To find the -intercept:
Step 1 - let
Step 2 - Solve for .
Step 3 - Write the intercept as a point.