*y*-intercept

In algebra class, we learned that the y-intercept of a graph is the value on the *y*-axis when *x* = 0. That is easy to find using the formula for the line, because we only have to plug in *x* = 0 and find *y*.

Since [latex]F=1.8\cdot{C}+32[/latex], when we plug in* C* = 0, then [latex]F=1.8\cdot{C}+32=1.8\cdot0+32=32[/latex]

When we look at the graph for this formula, go to *C* = 0, and find the point on the graph above that, the *y*-value appears to be about a little bit above 30. To find a more precise estimate from the graph, we’d need a bigger graph. This is consistent with the value of 32 that we found algebraically. Understanding both of these provides a way of checking your work on either of them.

## The formula for the line

Notice that both the slope and the *y*-intercept appear in the formula for the line. In algebra class, you learned that

- Any equation in which
*x*and*y*appear only to the first power is a linear equation. - The equation of a line can be written in a variety of ways, but if you take any of those and “solve for
*y*” you’ll get a formula like [latex]y=mx+b[/latex], where*y*is the output variable,*x*is the input variable, and*m*and*b*are numbers. Then the slope is the coefficient of the*x*-variable and the*y*-intercept is the constant. So*m*is the slope and*b*is the*y*-intercept.

For the temperature example, since *F* is the output value and *C* is the input value, and [latex]F=1.8\cdot{C}+32[/latex], then we see from the formula that the slope is 1.8 and the *y*-intercept is 32.** **

### Interpret the slope and intercept

- The
*y*-intercept is the value for*y*when - When the
*x*-value increases by 1 unit, then the*y*-value increases by the value of the slope.

So, for the temperature problem, we found that if the temperature was 0° C, then it is 32° F. And we also found that, if the temperature C increases by 1° C, then the temperature F increases by 1.8° F.

**Example 1**. For this formula: [latex]R=-6.3+8.2\cdot{d}[/latex], tell

- Which variable is the output variable?
- Which variable is the input variable?
- Is it linear relationship?
- If it is a linear relationship, what is the slope?
- If it is a linear relationship, what is the
*y*-intercept?

**Example 2**. For this formula: [latex]R=-6.3+8.2\cdot{d}[/latex], find three points that fit this and use them to sketch a graph.

**Example 3**. Determine whether these three points lie on a straight line by computing slopes: (3,5) and (1,4) and (4,6).