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Chapter 2: Problem 88

Write a rule for a linear function \(y=g(x)\), given that \(g(0)=7\) and \(g(-2)=4\).

Short Answer

Expert verified
The linear function is \(y = g(x) = \frac{3}{2}x + 7\).

Step by step solution

01

Identify the Key Information

We know two points on the line: \(0, 7\) and \(-2, 4\). These points will help in determining the slope and y-intercept of the line.
02

Calculate the Slope

Use the slope formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). Substitute the points \(0, 7\) and \(-2, 4\): \[m = \frac{7 - 4}{0 + 2} = \frac{3}{2}\].
03

Determine the Y-intercept

The y-intercept \(b\) is given by \(g(0)=7\). Therefore, \(b = 7\).
04

Write the Equation

Now that we have the slope \(m = \frac{3}{2}\) and y-intercept \(b = 7\), we can write the equation of the line: \[y = g(x) = \frac{3}{2}x + 7\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope formula
The slope of a line represents how steep it is. In mathematical terms, the slope is often denoted by the letter \(m\) and tells us how much \(y\) changes for a change in \(x\).

To find the slope between two points, we use the slope formula:
\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]
This formula means you subtract the y-values and then divide by the difference of the x-values. In our example, we have the points (0, 7) and (-2, 4). Plugging these values into the slope formula, we get:
\[m = \frac{7 - 4}{0 + 2} = \frac{3}{2}\]
The slope \(m\) is \(\frac{3}{2}\). This means for every 2 units we move horizontally, the line goes up by 3 units.
y-intercept
The y-intercept is the point where the line crosses the y-axis. This is the value of \(y\) when \(x\) is zero.

For linear functions, the y-intercept is represented by the letter \(b\). In the equation \(y = mx + b\), \(b\) is the y-intercept.

In our exercise, we know that \(g(0) = 7\). This means when \(x\) is 0, \(y\) is 7. So, the y-intercept \(b\) is 7.
It's a key point used to graph or understand the line equation more effectively.
equation of a line
The equation of a line describes the relationship between the \(x\) and \(y\) coordinates on the line. For a linear function, this equation is generally written in the form \(y = mx + b\).

Here,
  • \(m\) is the slope
  • \(b\) is the y-intercept
From our previous sections, we've determined that \(m = \frac{3}{2}\) and \(b = 7\).
By substituting these values into the general form \(y = mx + b\), we get:
\[y = \frac{3}{2}x + 7\]
This is the equation of the line that passes through the points (0, 7) and (-2, 4).
It means no matter what \(x\) value you plug into this equation, the equation will tell you the corresponding \(y\) value on the line.

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Most popular questions from this chapter

While on vacation in France, Sadie bought a box of almond croissants. Each croissant cost \(鈧� 2.4\) (euros). a. Write a function that represents the \(\operatorname{cost} C(x)\) (in euros) for \(x\) croissants. b. At the time of the purchase, the exchange rate was \(\$ 1=鈧� 0.80 .\) Write a function that represents the amount \(D(C)\) (in \$) for \(C\) euros spent. c. Evaluate \((D \circ C)(x)\) and interpret the meaning in the context of this problem. d. Evaluate \((D \circ C)(12)\) and interpret the meaning in the context of this problem.

Refer to the functions \(r, p,\) and \(q .\) Evaluate the function and write the domain in interval notation. \(r(x)=-3 x \quad p(x)=x^{2}+3 x \quad q(x)=\sqrt{1-x}\) $$(r-p)(x)$$

Evaluate the step function defined by \(f(x)=[x]\) for the given values of \(x\). $$ f(0.09) $$

A function is given. (See Examples \(4-5)\) a. Find \(f(x+h)\). b. Find \(\frac{f(x+h)-f(x)}{h}\). $$f(x)=x^{2}-3 x$$

Use a graphing utility to graph the piecewise-defined function. $$ \begin{aligned} &z(x)=\left\\{\begin{array}{ll} 2.5 x+8 & \text { for } x<-2 \\ -2 x^{2}+x+4 & \text { for }-2 \leq x<2 \\ -2 & \text { for } x \geq 2 \end{array}\right.\\\ &\text { Is there actually a "gap" in the graph at } x=2 ? \end{aligned} $$
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