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Chapter 0: Problem 75

Writing Functions, write an equation for a function that has the given graph. Line segment connecting \((-4,3)\) and \((0,-5)\)

Short Answer

Expert verified
The function for the given graph of the line segment is \( y = -2x - 5\).

Step by step solution

01

Calculate the Slope

First, find the slope of the line segment. The slope is given by the formula \((y2-y1) / (x2-x1)\). Using the two points provided \((-4,3)\) and \((0,-5)\), therefore, the slope \(m\) becomes \((-5-3)/(0-(-4)) = -2\).
02

Find the y-intercept

With the calculated slope and the given points, the value of the y-intercept can be calculated using the formula \(b = y - mx\). Using one of the points \((0,-5)\) and slope of -2, we can calculate \(b = -5 - (-2*0) = -5\)
03

Write the Equation

Now we have both the slope \(m = -2\) and the y-intercept \(b = -5\). Write these into the line formula \(y = mx + b\) to get the final equation \(y = -2x - 5\)

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Key Concepts

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

Slope Calculation
The slope of a line is a measure that describes its steepness. It tells us how much the line rises or falls as we move from left to right.
Think of it as the rate of change, similar to how speed measures how fast something moves. To find the slope, we start by taking two points on the line. In our case, these points are
  • Point A: \((-4,3\))
  • Point B: \((0,-5\))
The formula for calculating slope \(m\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging the points into the formula, \[ m = \frac{-5 - 3}{0 - (-4)} = \frac{-8}{4} = -2 \]
This tells us the slope of the line is \(-2\), which means for every unit we move to the right, the line goes down by 2 units.
Y-intercept
Once we have the slope, the next step is to find the y-intercept. The y-intercept is the point where the line crosses the y-axis. In a graph, it's where \(x = 0\). Thus, we only need the y-value of that point to find the y-intercept.
We use the formula for a line, which is \[ y = mx + b \] Where:
  • \(m\) is the slope
  • \(b\) is the y-intercept
For our line, using point \((0, -5)\), the y-coordinate is \(-5\). Since this is where the line crosses the y-axis, the \(y\)-intercept is simply \(-5\). Now, you can often find \(b\) using the slope formula again with a known point and the calculated slope, but here it is straightforward, as we directly see \(b = -5\) when \(x = 0\). This will be a key component when writing the equation of the line.
Equation of a Line
Creating the equation of a line involves assembling the elements we found: the slope and y-intercept. The standard formula for the equation of a line is \[ y = mx + b \] This equation is simple but very powerful in representing straight lines. Here,
  • \(m\) (slope) is \(-2\)
  • \(b\) (y-intercept) is \(-5\)
Now, using these values, write the equation of our line: \[ y = -2x - 5 \] This tells us that for any \(x\)-value, you can plug it into this equation to find the corresponding \(y\)-value on the line. The negative slope means the line is decreasing, moving downwards as it goes from left to right. This equation can now be used to plot or understand the behavior of the line visually.

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Most popular questions from this chapter

Find the composite functions \((f \circ g)\) and \((g \circ f)\). What is the domain of each composite function? Are the two composite functions equal? \(f(x)=x^{2}-1\) \(g(x)=\cos x\)

Sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. $$h(x)=\sqrt{x-1}$$

Car Performance A V8 car engine is coupled to a dynamometer and the horsepower \(y\) is measured at different engine speeds \(x\) (in thousands of revolutions per minute). The results are shown in the table. $$ \begin{array}{|c|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 40 & 85 & 140 & 200 & 225 & 245 \\ \hline \end{array} $$ (a) Use the regression capabilities of a graphing utility to find a cubic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the model to approximate the horsepower when the engine is running at 4500 revolutions per minute.

Find the composite functions \((f \circ g)\) and \((g \circ f)\). What is the domain of each composite function? Are the two composite functions equal? \(f(x)=\frac{3}{x}\) \(g(x)=x^{2}-1\)

Volume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) Write the volume \(V\) as a function of \(x\), the length of the corner squares. What is the domain of the function? (b) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume. (c) Use the table feature of a graphing utility to verify your answer in part (b). (The first two rows of the table are shown.) $$ \begin{array}{|c|c|c|} \hline \text { Height, } x & \begin{array}{c} \text { Length } \\ \text { and Width } \end{array} & \text { Volume, } \boldsymbol{V} \\ \hline 1 & 24-2(1) & 1[24-2(1)]^{2}=484 \\ \hline 2 & 24-2(2) & 2[24-2(2)]^{2}=800 \\ \hline \end{array} $$
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