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Chapter 1: Problem 22

Find an equation of the line that passes through the points, and sketch the line. $$ (1,-2),(3,-2) $$

Short Answer

Expert verified
The equation of the line that passes through the points (1,-2) and (3,-2) is \(y = -2\).

Step by step solution

01

Determine the Slope

The slope (m) of a line is given by the change in y-coordinates divided by the change in x-coordinates. Since the y-coordinates of both points are the same (-2), the slope of the line is zero. Formally, it's calculated as \(m =\frac{(y_2 - y_1)}{(x_2 - x_1)}=\frac{(-2) - (-2)}{(3 - 1)} = 0\).
02

Use point-slope form to obtain the line equation

The point-slope form of a line equation is \(y - y_1 = m(x - x_1)\). Substituting values, \(y - (-2) = 0(x -1)\), and after simplifying this equation, the result is \(y = -2\).
03

Sketching the line

Since the slope is zero, the line is horizontal and passes through the points (1,-2) and (3,-2). It is a horizontal line at y = -2.

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Most popular questions from this chapter

Determine conditions on the constants \(a, b,\) and \(c\) such that the graph of \(f(x)=\frac{a x+b}{c x-a}\) is symmetric about the line \(y=x\).

(a) Let \(f_{1}(x)\) and \(f_{2}(x)\) be continuous on the closed interval \([a, b]\). If \(f_{1}(a)f_{2}(b),\) prove that there exists \(c\) between \(a\) and \(b\) such that \(f_{1}(c)=f_{2}(c)\). (b) Show that there exists \(c\) in \(\left[0, \frac{\pi}{2}\right]\) such that \(\cos x=x\). Use a graphing utility to approximate \(c\) to three decimal places.

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}+x-1, \quad[0,5], \quad f(c)=11 $$

Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arctan x+\frac{\pi}{2} $$

Writing Use a graphing utility to graph \(f(x)=x, \quad g(x)=\sin x, \quad\) and \(\quad h(x)=\frac{\sin x}{x}\) in the same viewing window. Compare the magnitudes of \(f(x)\) and \(g(x)\) when \(x\) is "close to" \(0 .\) Use the comparison to write \(a\) short paragraph explaining why \(\lim _{x \rightarrow 0} h(x)=1\).
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