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Chapter 2: Problem 33

If you know a point on a line and you know the equation of a line perpendicular to this line, explain how to write the line's equation.

Short Answer

Expert verified
The equation of the desired line can be formed by following these steps: 1) Determine the slope of the provided line, 2) Calculate the negative reciprocal of that slope, and 3) Use the point-slope formula to create the equation of the line that passes through the given point and has the computed slope.

Step by step solution

01

Find the slope of the provided perpendicular line

Take the equation of the provided line which is perpendicular to the desired line. Our task is to identify its slope, generally denoted as \(m\). If the equation is given in slope-intercept form \(y = mx + b\), then the slope directly corresponds to \(m\). If it isn't, initially convert the equation into slope-intercept form. Achieve this by arranging terms so that \(y\) is isolated on the left side, \(y = mx + b\), where \(m\) will be the coefficient of \(x\) on the right.
02

Calculate the slope of the required line

Since we've identified the slope of the provided line, we can now find the slope of the desired line, which is the negative reciprocal of the slope of the provided line. On paper, if the slope of the given line is \(m\), then the slope of the perpendicular line would be \(-1/m\). Take the reciprocal of the provided line's slope (\(1/m\)) and change the sign (to make it negative or positive, respectively).
03

Apply the point-slope formula

We now know a fixed point through which our required line passes and we also know its slope. Using these two pieces of information, we can write the equation of the required line. As per the point-slope formula, if a line with a slope of \(m\) passes through the point \((x1, y1)\), the equation of the line is \(y - y1 = m(x - x1)\). Substitute the known point and calculated slope into this formula to derive the equation of the required line.

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Most popular questions from this chapter

Here is the 2011 Federal Tax Rate Schedule \(X\) that specifies the tax owed by a single taxpayer. (TABLE CAN'T COPY) The preceding tax table can be modeled by a piecewise function, where \(x\) represents the taxable income of a single taxpayer and \(T(x)\) is the tax owed: $$T(x)=\left\\{\begin{array}{c}0.10 x \\\850.00+0.15(x-8500) \\\4750.00+0.25(x-34,500) \\\17,025.00+0.28(x-83,600) \\\\\frac{?}{?}\end{array}\right.$$ if \(\quad 0 < x \leq 8500\) if \(\quad 8500 < x \leq 34,500\) if \(\quad 34,500 < x \approx 83,600\) if \(\quad 83,600 < x =174,400\) if \(174,400 < x \leq 379,150\) if \(\quad x >379,150\) Use this information to solve. Find and interpret \(T(50,000)\).

Suppose that \(h(x)=\frac{f(x)}{g(x)} .\) The function \(f\) can be even, odd, or neither. The same is true for the function \(g .\) a. Under what conditions is \(h\) definitely an even function? b. Under what conditions is \(h\) definitely an odd function?

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Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ g(x)-(x-3)^{3} $$

If you are given a function's equation, how do you determine if the function is even, odd, or neither?
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