G1.02: Intercepts & Example 3

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

  1. Any equation in which x and y appear only to the first power is a linear equation.
  2. 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

  1. The y-intercept is the value for y when
  2. 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

  1. Which variable is the output variable?
  2. Which variable is the input variable?
  3. Is it linear relationship?
  4. If it is a linear relationship, what is the slope?
  5. 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).