2.1.1: Exercises 2.1

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Terms and Concepts

Exercise \(\PageIndex{1}\)

Explain the difference between point-slope form and slope-intercept form.

Answer: Answers will vary.

Exercise \(\PageIndex{2}\)

To uniquely determine a line, what information do you need?

Answer: A point and a slope, or two points

Exercise \(\PageIndex{3}\)

What is the slope of a horizontal line?

Answer: \(m=0\)

Exercise \(\PageIndex{4}\)

A line goes through the point \((0,6)\). Is this the y-intercept of the line or the x-intercept of the line? Explain.

Answer: This is the y-intercept because \(x=0\) so it is where the line crosses the y-axis.

Exercise \(\PageIndex{5}\)

Line 1 has a slope of \(m_1=2\). If line 2 is parallel to line 1, what is \(m_2\)?

Answer: \(m_2=2\)

Exercise \(\PageIndex{6}\)

Line 1 has a slope of \(m_1=-4\). If line 2 is perpendicular to line 1, what is \(m_2\)?

Answer: \(m_2=\frac{1}{4}\)

Problems

In exercises \(\PageIndex{7}\) - \(\PageIndex{16}\), write an equation for each line in the indicated form.

Exercise \(\PageIndex{7}\)

Write the equation in point-slope form for the line that passes through \((1,2)\) and is parallel to the line \(2x+y=5\).

Answer: \(y-2=-2(x-1)\)

Exercise \(\PageIndex{8}\)

Write the equation of the line in slope-intercept form passing through the points \((1,2)\) and \((-1,4)\).

Answer: \(y=-x+3\)

Exercise \(\PageIndex{9}\)

Write the equation in point-slope form for the line that passes through \((0,4)\) and is perpendicular to the line \(x-2y=6\).

Answer: \(y-4=-2(x-0)\)

Exercise \(\PageIndex{10}\)

Write the equation of the line in slope-intercept form passing through the points \((-1,0)\) and \((3,6)\).

Answer: \(y=\frac{3}{2}x + \frac{3}{2}\)

Exercise \(\PageIndex{11}\)

Write the equation of the line in slope-intercept form passing through the points \((-2,1)\) and \((2,7)\).

Answer: \(y=\frac{3}{2}x + 4\)

Exercise \(\PageIndex{12}\)

Consider the linear function \(f(x)=2x-8\). What is the value of the function when \(x=0.1\)?

Answer: \(-7.8\)

Exercise \(\PageIndex{13}\)

Write the equation in slope-intercept form for the line that passes through \((-2,2)\) and is perpendicular to the line \(x+3y=8\).

Answer: \(y=3x+8\)

Exercise \(\PageIndex{14}\)

Write the equation in point-slope form of the line that passing through the points \((3,6)\) and \((7,4)\).

Answer: \(y-4=\frac{1}{2}(x-7)\), or \(y-6=-\frac{1}{2}(x-3)\)

Exercise \(\PageIndex{15}\)

Write the equation of the line passing through the points \((-4,4)\) and \((0,-4)\) in slope-intercept form.

Answer: \(y=-2x-4\)

Exercise \(\PageIndex{16}\)

Write the equation of the line parallel to \(y=6x+4\) that has a y-intercept of \(2\) in point-slope form.

Answer: \(y-2=6(x-0)\)

In exercises \(\PageIndex{17}\) - \(\PageIndex{20}\), answer each question about the properties of the given line(s).

Exercise \(\PageIndex{17}\)

Consider the linear function \(g(x)=-4x+5\). What is the slope of the function when \(x=4\)?

Answer: \(m=-4\)

Exercise \(\PageIndex{18}\)

Determine the x-intercept of the line \(y=4x-8\).

Answer: \((2,0)\)

Exercise \(\PageIndex{19}\)

Determine the y-intercept of the line \(y=4x-8\).

Answer: \((0,-8)\)

Exercise \(\PageIndex{20}\)

Which line has a steeper slope: \(y=5x+10\) or the line passing through the points \((-5,0)\) and \((0,11)\)?

Answer: The line \(y=5x+10\) has a steeper slope.