Slope & Rate of Change 100
Find the slope of the line through the points (1, 3) and (4, 9). 200
slope = 2
Write the equation of a line in slope-intercept form with slope 4 and y-intercept −2
y=4x−2
Sketch (or describe) the line through (0,0) and (2,2). What is the slope?
Line through (0,0) and (2,2): equation y=x. Slope = 1.
What is the slope of any line parallel to y=−3x+7y=−3x+7?
Any line parallel to y=−3x+7 has slope −3.
A taxi company charges a $3 flat fee plus $2.50 per mile. Write the linear equation for cost C in terms of miles m.
Taxi cost: C(m)=3+2.50m
A line has slope 2/3. If x increases by 9, by how much does y change?
change in y = slope × change in x = (2/3)×9=6
Convert the equation 3x+6y=12 to slope-intercept form.
3x+6y=12 ⇒ 6y=−3x+12 ⇒ y=−(1/2)x+2
Find the equation of the line through (−1, 4) and (3, 4).
Points (−1,4) and (3,4): horizontal line y=4.
Give the slope of a line perpendicular to y=(4/5)x−1.
A perpendicular slope to y=(4/5)x−1 is the negative reciprocal: −5/4.
A phone plan charges a monthly fee of $20 and $0.05 per text. Express the total monthly cost T as a function of the number of texts t, and find the cost for 300 texts.
For 300 texts: T(300)=20+0.05(300)=20+15=35
Determine the slope from the graph: a line passes through (−2, 5) and (3, −10).
slope = −3
Write the equation of a line in point-slope form that goes through (2, −1) with slope −1/2.
Point-slope: y−(−1) = −(1/2)(x−2) or y+1=−(1/2)(x−2)
Find the equation of the line through (5, −2) and (−1, 4).
y=−x+3
Find the equation of a line through (0, −2) that is perpendicular to the line 2x−3y=6.
y=−(2/3)x−2
A line models temperature T (°F) over time t (hours): T=−2t+86. What is the temperature after 4 hours? Interpret the slope practically.
T(t)=−2t+86 ⇒ T(4)=−2(4)+86=78∘F.
Slope −2 means temperature falls 2°F each hour.
The table shows values of y for x: (0, 2), (1, 5), (2, 8). Is the rate of change constant? If so, find it and interpret what it means here.
Yes. Rate of change = 3. Interpretation: y increases by 3 for each 1 increase in x
Given the graph of a vertical line through x = 7, write its equation. Explain why it cannot be written in slope-intercept form.
Equation: x=7. Vertical lines have undefined slope, so they cannot be written in slope-intercept form y=mx+b.
Are the points (1,2), (2,5), and (3,8) collinear (on the same line)? Explain and, if yes, give the equation of the line.
Points are collinear. Equation: y=3x−1
Are the lines y=2x+1 and 4x−2y=7 parallel, perpendicular, or neither?
they are parallel
Lena walks from point A at (0,0) to point B at (3, 4) on flat terrain. If her path is modeled by a straight line, find that line's equation and explain the meaning of its slope in this context.
Equation: y= (4/3)x.
Slope means 4 units north (rise) for every 3 units east (run).
A line has equation y=mx+b. If the average rate of change between x = 2 and x = 5 is 6, what is m? Explain.
For a linear function the slope equals the average rate of change, so m=6.
Put the line y−3=2(x+1) into standard form Ax+By=C with integer A ≥ 0 and gcd(A,B,C)=1.
−2x+y=−5
2x−y=5
Given two points with coordinates (a, 2a) and (3a, 8) where a ≠ 0, find the value(s) of a for which these points lie on a line of slope 3. Then give the equation of that line.
For a=1 the points are (1,2) and (3,8); equation: y=3x−1.
Find the equation of the line passing through (6, 1) that is parallel to the line through (2, −3) and (5, 0).
y=x−5
Two companies have linear cost models. Company A: CA(x)=50+8x. Company B: CB(x)=20+10x. For what x (number of items) do costs equal? Which company is cheaper for x is less than that value and for x is greater than that value?
x=15
At x<15, Company B is cheaper; at x>15, Company A is cheaper.