Graphs

The equations of two lines are given. Determine if the lines are parallel, perpendicular, or neither. $y=-2x+3$, $y=-\frac{1}{2}x+2$

The equations of two lines are given. Determine if the lines are parallel, perpendicular, or neither. $y=\frac{1}{2}x-3$, $y=-2x+4$

The equations of two lines are given. Determine if the lines are parallel, perpendicular, or neither. $y=4x+5$, $y=-4x+2$

The equations of two lines are given. Determine if the lines are parallel, perpendicular, or neither. $y=2x-3$, $y=2x+4$

Write an equation of each line. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer.

Write an equation of each line. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer.

Write an equation of each line. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer.

Write an equation of each line. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer.

Use slopes to show that the triangle whose

vertices are $(-2,5), (1,3)$ and $(-1,0)$ is a right triangle.

Use slopes to show that the quadrilateral whose vertices are $(1,-1), (4,1), (2,2)$ and $(5,4)$ is a parallelogram.

Use slopes to show that the quadrilateral

whose vertices are $(-1,0), (2,3), (1,-2)$ and $(4,1)$ is a rectangle.

Use slopes and the distance formula to show that the quadrilateral whose vertices are $(0,0), (1,3), (4,2)$ and $(3,-1)$ is a square.

A truck rental company rents a moving truck for one day by charging \(29 plus \)0.20 per mile. Write a linear equation that relates the cost C, in dollars, of renting the truck to the number x of miles driven. What is the cost of renting the truck if the truck is driven 110 miles? 230 miles?

The fixed costs of operating a business are the costs incurred regardless of the level of production. Fixed costs include rent, fixed salaries, and costs of leasing machinery. The variable costs of operating a business are the costs that change with the level of output. Variable costs include raw materials, hourly wages, and electricity. Suppose that a manufacturer of jeans has fixed daily costs of \(500 and variable costs of \)8 for each pair of jeans manufactured. Write a linear equation that relates the daily cost C, in dollars, of manufacturing the jeans to the number x of jeans manufactured. What is the cost of manufacturing 400 pairs of jeans? 740 pairs?

The annual fixed costs for owning a small sedan are \(1289, assuming the car is completely paid for. The cost to drive the car is approximately \)0.15 per mile. Write a linear equation that relates the cost C and the number x of miles driven annually.

Dan receives $375 per week for selling new and used cars at a car dealership in Oak Lawn, Illinois. In addition, he receives 5% of the profit on any sales that he generates. Write a linear equation that represents Dan’s weekly salary S when he has sales that generate a profit of x dollars.

Commonwealth Edison Company supplies electricity to residential customers for a monthly customer charge of $11.47 plus 11 cents per kilowatt-hour for up to 600 kilowatt-hours.

(a) Write a linear equation that relates the monthly charge C, in dollars, to the number x of kilowatt-hours used in a month, $0\le x\le 600$

(b) Graph this equation.

(c) What is the monthly charge for using 200 kilowatthours?

(d) What is the monthly charge for using 500 kilowatthours?

(e) Interpret the slope of the line.

Florida Power & Light Company supplies electricity to residential customers for a monthly customer charge of $5.90 plus 8.81 cents per kilowatt-hour for up to 1000 kilowatt-hours.

(a) Write a linear equation that relates the monthly charge C, in dollars, to the number x of kilowatt hours used in a month, $0\le x\le 1000$

(b) Graph this equation.

(c) What is the monthly charge for using 200 kilowatthours?

(d) What is the monthly charge for using 500 kilowatthours?

(e) Interpret the slope of the line.

The relationship between Celsius (°C) and Fahrenheit (°F) degrees of measuring temperature is linear. Find a linear equation relating °C and °F if 0°C corresponds to 32°F and 100°C corresponds to 212°F. Use the equation to find the Celsius measure of 70°F.

The Kelvin (K) scale for measuring temperature is obtained by adding 273 to the Celsius temperature.

(a) Write a linear equation relating K and °C.

(b) Write a linear equation relating K and °F.

A wooden access ramp is being built to reach a platform that sits 30 inches above the floor. The ramp drops 2 inches for every 25-inch run.

(a) Write a linear equation that relates the height y of the ramp above the floor to the horizontal distance x from the platform.

(b) Find and interpret the x-intercept of the graph of your equation.

(c) Design requirements stipulate that the maximum run be 30 feet and that the maximum slope be a drop of 1 inch for each 12 inches of run. Will this ramp meet the requirements? Explain.

(d) What slopes could be used to obtain the 30-inch rise and still meet design requirements?

A report in the Child Trends DataBase indicated that, in 2000, 20.6% of twelfth grade students reported daily use of cigarettes. In 2009, 11.2% of twelfth grade students reported daily use of cigarettes.

(a) Write a linear equation that relates the percent y of twelfth grade students who smoke cigarettes daily to the number x of years after 2000.

(b) Find the intercepts of the graph of your equation.

(c) Do the intercepts have any meaningful interpretation?

(d) Use your equation to predict the percent for the year 2025. Is this result reasonable?

A cereal company finds that the number of people who will buy one of its products in the first month that it is introduced is linearly related to the amount of money it spends on advertising. If it spends \(40,000 on advertising, then 100,000 boxes of cereal will be sold, and if it spends \)60,000, then 200,000 boxes will be sold.

(a) Write a linear equation that relates the amount A spent on advertising to the number x of boxes the company aims to sell.

(b) How much advertising is needed to sell 300,000 boxes of cereal?

(c) Interpret the slope.

Show that the line containing the points (a, b) and (b, a), a  b, is perpendicular to the line y = x. Also show that the midpoint of (a, b) and (b, a) lies on the line y = x.

The equation $2x-y=C$ defines a family of lines, one line for each value of $C$. On one set of coordinate axes, graph the members of the family when $C=-4,C=0$ and $C=2$. Can you draw a conclusion from the graph about each member of the family?

Prove that if two nonvertical lines have slopes whose product is -1 then the lines are perpendicular.

The accepted symbol used to denote the slope of a line is the letter m. Investigate the origin of this symbolism. Begin by consulting a French dictionary and looking up the French word Monter. Write a brief essay on your findings.

The term grade is used to describe the inclination of a road. How does this term relate to the notion of slope of a line? Is a 4% grade very steep? Investigate the grades of some mountainous roads and determine their slopes. Write a brief essay on your findings.

Which of the following equations might have the graph shown? (More than one answer is possible.)

$$\begin{gathered} \left(a\right) 2x + 3y = 6 \\ \left(b\right) -2x + 3y = 6 \\ \left(c\right) 3x - 4y = -12 \\ \left(d\right) x - y = 1 \\ \left(e\right) x - y = -1 \\ \left(f\right) y = 3x - 5 \\ \left(g\right) y = 2x + 3 \\ \left(h\right) y = -3x + 3 \end{gathered}$$

Which of the following equations might have the graph shown? (More than one answer is possible.)

$$\begin{gathered} \left(a\right) 2x + 3y = 6 \\ \left(b\right) 2x - 3y = 6 \\ \left(c\right) 3x + 4y = 12 \\ \left(d\right) x - y = 1 \\ \left(e\right) x - y = -1 \\ \left(f\right) y = -2x - 1 \\ \left(g\right) y = - \frac{1}{2} x + 10 \\ \left(h\right) y = x + 4 \end{gathered}$$

The figure shows the graph of two parallel lines. Which of the following pairs of equations might have such a graph?

$$\begin{gathered} \left(a\right) x - 2y = 3 \\ x + 2y = 7 \\ \left(b\right) x + y = 2 \\ x + y = -1 \\ \left(c\right) x - y = -2 \\ x - y = 1 \\ \left(d\right) x - y = -2 \\ 2x - 2y = -4 \\ \left(e\right) x + 2y = 2 \\ x + 2y = -1 \end{gathered}$$

The figure shows the graph of two perpendicular lines. Which of the following pairs of equations might have such a graph?

$$\begin{gathered} \left(a\right) y - 2x = 2 \\ y + 2x = -1 \\ \left(b\right) y - 2x = 0 \\ 2y + x = 0 \\ \left(c\right) 2y - x = 2 \\ 2y + x = -2 \\ \left(d\right) y - 2x = 2 \\ x + 2y = -1 \\ \left(e\right) 2x + y = -2 \\ 2y + x = -2 \end{gathered}$$

Carpenters use the term pitch to describe the steepness of staircases and roofs. How does pitch relate to slope? Investigate typical pitches used for stairs and for roofs. Write a brief essay on your findings.

Can the equation of every line be written in slope–intercept form? Why?

Does every line have exactly one x intercept and one y-intercept? Are there any lines that have no intercepts?

What can you say about two lines that have equal slopes and equal y-intercepts?

What can you say about two lines with the same $x$-intercept and the same $y$-intercept Assume that the $x$-intercept is not 0.

If two distinct lines have the same slope, but different $x$-intercepts, can they have the same $y$-intercept?

If two distinct lines have the same $y$

intercept, but different slopes, can they have the same $x$-intercept?

Which form of the equation of a line do you prefer to use? Justify your position with an example that shows that your choice is better than another. Have reasons.

A student is asked to find the slope of the line joining (-3, 2) and (1, -4). He states that the slope is $\frac{3}{2}$. Is he correct? If not, what went wrong?

Open the slope applet. Move point B around the Cartesian plane with your mouse.

(a) Move B to the point whose coordinates are ( 2, 7). What is the slope of the line?

(b) Move B to the point whose coordinates are ( 3, 6). What is the slope of the line?

(c) Move B to the point whose coordinates are (4, 5). What is the slope of the line?

(d) Move B to the point whose coordinates are (4, 4). What is the slope of the line?

(e) Move B to the point whose coordinates are (4, 1). What is the slope of the line?

(f) Move B to the point whose coordinates are (3, -2). What is the slope of the line?

(g) Slowly move B to a point whose x-coordinate is 1. What happens to the value of the slope as the x-coordinate approaches 1?

(h) What can be said about a line whose slope is positive? What can be said about a line whose slope is negative? What can be said about a line whose slope is 0?

(i) Consider the results of parts (a)–(c). What can be said about the steepness of a line with positive slope as its slope increases?

( j) Move B to the point whose coordinates are (3, 5). What is the slope of the line? Move B to the point whose coordinates are (5, 6). What is the slope of the line? Move B to the point whose coordinates are (-1, 3). What is the slope of the line?

To complete the square of $x^2+10x$, you would (add/subtract) the number ______. 

Use the Square Root Method to solve the equation ${\left(x-2\right)}^2=9$

True or False 

Every equation of the form

$x^2+y^2+ax+by+c=0$ has a circle as its graph.

For a circle, the ______ is the distance from the center to any point on the circle.

True or False 

The radius of the circle $x^2+y^2=9$ is $3$.

The center of the circle ${\left(x+3\right)}^2+{\left(y-2\right)}^2=13$ is $\left(3,-2\right)$

Find the center and radius of the circle. Write the standard form of the equation.

Find the center and radius of the circle. Write the standard form of the equation.