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- 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.
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.
- 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?
Answer:
- The output variable is R.
- the input variable is d.
- yes, it is a linear relationship because both variables appear only to the first power.
- the slope is 8.2 because that is the number multiplied by the input variable.
- the y-intercept is -6.3.
Answer: To find a point that fits this relationship, choose any value for the input variable. (Generally we choose a value that is easy to compute with and to graph, such as a small whole number.) Let’s begin by choosing [latex]d=2[/latex]. Then compute the output value. [latex-display]\begin{align}&R=-6.3+8.2\cdot{d}\\&R=-6.3+8.2\cdot(2)\\&R=-6.3+16.4\\&R=10.1\\\end{align}[/latex-display] Now choose two other values for d and use them to compute the output values. In the interests of saving space, we will omit the details. Choose [latex]d=5[/latex] and find that [latex]R=-6.3+8.2\cdot5=34.7[/latex] Choose [latex]d=7[/latex] and find that [latex]R=-6.3+8.2\cdot7=51.1[/latex]
| Graph with unconnected points | Graph with points connected by a line |
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Answer: Find the slope of the line through (3,5) and (1,4). [latex]m=\frac{rise}{run}=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{5-4}{3-1}=\frac{1}{2}=0.5[/latex]
Find the slope of the line through (3,5) and (4,6). [latex]m=\frac{rise}{run}=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{5-6}{3-4}=\frac{-1}{-1}=1.0[/latex]
Since these two slopes aren’t equal, these three points are not on a straight line.
Check: Check this by graphing the three points to see if they appear to lie on a straight line. In the following graph, we have drawn the line between the points and it is clear that it is not the same straight line between all three points.
Licenses & Attributions
CC licensed content, Shared previously
- Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution.
