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Chapter 1: Problem 30

Plot the points and find the slope of the line passing through the pair of points. $$(-2,1),(-4,-5)$$

Short Answer

Expert verified
The slope of the line passing through the points (-2,1) and (-4,-5) is 3.

Step by step solution

01

Plotting the Points

Firstly, place the two points (-2,1) and (-4,-5) on a graph. The first coordinate describes the position on the x-axis, while the second shows the position on the y-axis. For example, the point (-2,1) means moving 2 units to the left from the center (since it's negative) and 1 unit up (since it's positive).
02

Joining the Points

Once the points are marked, draw a line passing through both points. This line's slope is what needs to be calculated.
03

Calculating the Slope

Using the slope formula, which is \(slope (m) = \frac{(y2-y1)}{(x2-x1)} \), where (x1,y1) and (x2,y2) are coordinates of the given points, plug in the values of the points we have. So, plugging in the given points (-2,1) and (-4,-5) into the formula, the calculation becomes: \(slope (m) = \frac{(-5-1)}{(-4-(-2))}\). When you calculate that, it simplifies to \( m = \frac{-6}{-2}\).
04

Finalizing the Slope

After simplifying the last step, we end up with the slope (m)=3. This is the final slope of the line passing through the points (-2,1) and (-4,-5).

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Most popular questions from this chapter

Graph each of the functions with a graphing utility. Determine whether the function is even, odd, or neither. $$\begin{aligned}&\begin{array}{ll}f(x)=x^{2}-x^{4} & g(x)=2 x^{3}+1 \\\h(x)=x^{5}-2 x^{3}+x & j(x)=2-x^{6}-x^{8}\end{array}\\\&k(x)=x^{5}-2 x^{4}+x-2 \quad p(x)=x^{9}+3 x^{5}-x^{3}+x \end{aligned}$$

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) Use the fact that 13 inches is approximately the same length as 33 centimeters to find a mathematical model that relates centimeters \(y\) to inches \(x\). Then use the model to find the numbers of centimeters in 10 inches and 20 inches.

(a) use a graphing utility to graph the function and (b) state the domain and range of the function. $$k(x)=4\left(\frac{1}{2} x-\left[\left[\frac{1}{2} x\right]\right]\right)^{2}$$

The table shows the numbers of tax returns (in millions) made through e-file from 2003 through \(2010 .\) Let \(f(t)\) represent the number of tax returns made through e-file in the year \(t .\) (Source: Internal Revenue Service) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number of Tax Returns Made Through E-File } \\\\\hline 2003 & 52.9 \\\2004 & 61.5 \\\2005 & 68.5 \\\2006 & 73.3 \\\2007 & 80.0 \\\2008 & 89.9 \\\2009 & 95.0 \\\2010 & 98.7 \\\\\hline\end{array}$$ (a) Find \(\frac{f(2010)-f(2003)}{2010-2003}\) and interpret the result in the context of the problem. (b) Make a scatter plot of the data. (c) Find a linear model for the data algebraically. Let \(N\) represent the number of tax returns made through e-file and let \(t=3\) correspond to 2003 (d) Use the model found in part (c) to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}\hline t & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline N & & & & & & & & \\ \hline\end{array}$$ (e) Compare your results from part (d) with the actual data. (f) Use a graphing utility to find a linear model for the data. Let \(x=3\) correspond to \(2003 .\) How does the model you found in part (c) compare with the model given by the graphing utility?

(a) Write the linear function \(f\) such that it has the indicated function values and (b) Sketch the graph of the function. $$f(-3)=-8, \quad f(1)=2$$
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