Chapter 4: Graphs of Linear Equations and Functions
Introduction
The ability to graph linear equations and functions is important in mathematics. In fact, graphing equations and solving equations are two of the most important concepts in mathematics. If you master these, all mathematical subjects will be much easier, even Calculus!
This chapter focuses on the visual representations of linear equations. You will learn how to graph lines from equations and write functions of graphed lines. You will also learn how to find the slope of a line and how to use a slope to interpret a graph.
Weather, such as temperature and the distance of a thunderstorm can be predicted using linear equations. You will learn about these applications and more in this chapter.
- 4.1.
Graphs in the Coordinate Plane
- 4.2.
Graphs of Linear Equations
- 4.3.
Horizontal and Vertical Line Graphs
- 4.4.
Applications of Linear Graphs
- 4.5.
Intercepts by Substitution
- 4.6.
Intercepts and the Cover-Up Method
- 4.7.
Slope
- 4.8.
Rates of Change
- 4.9.
Slope-Intercept Form
- 4.10.
Graphs Using Slope-Intercept Form
- 4.11.
Direct Variation
- 4.12.
Applications Using Direct Variation
- 4.13.
Function Notation and Linear Functions
- 4.14.
Graphs of Linear Functions
- 4.15.
Problem Solving with Linear Graphs
Chapter Summary
Summary
This chapter deals with graphing, including plotting points on a coordinate plane and graphing linear equations. As part of this topic, the chapter discusses horizontal and vertical graphs and analyzing linear graphs. It then moves on to intercepts and slopes, showing how to find intercepts by substituting and by using the Cover-Up Method and why slopes should be interpreted as rates of change. Slope-intercept form is also covered. Next, direct variation is talked about, and many real-world problems are solved. Finally, the chapter concludes by touching on function notation and linear functions, giving instruction on the graphs of linear functions and how to solve problems by making and reading a graph.
Graphing Linear Equations and Functions Review
Define the following words:
- \begin{align*}x-\end{align*}intercept
- \begin{align*}y-\end{align*}intercept
- direct variation
- parallel lines
- rate of change
In 6 – 11, identify the coordinates using the graph below.
- \begin{align*}D\end{align*}
- \begin{align*}F\end{align*}
- \begin{align*}A\end{align*}
- \begin{align*}E\end{align*}
- \begin{align*}B\end{align*}
- \begin{align*}C\end{align*}
In 12 – 16, graph the following ordered pairs on one Cartesian Plane.
- \begin{align*}\left (\frac{1}{2},-4\right )\end{align*}
- (–6, 1)
- (0, –5)
- (8, 0)
- \begin{align*}\left (\frac{3}{2},\frac{8}{4}\right )\end{align*}
In 17 and 18, graph the function using the table.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
–2 | 7 |
–1 | 9 |
0 | 11 |
1 | 13 |
2 | 15 |
- \begin{align*}& d && 0 && 1 && 2 && 3 && 4 && 5 && 6\\ & t && 0 && 75 && 150 && 225 && 300 && 375 && 450\end{align*}
In 19 – 24, graph the following lines on one set of axes.
- \begin{align*}y=-\frac{3}{2}\end{align*}
- \begin{align*}x=4\end{align*}
- \begin{align*}y=5\end{align*}
- \begin{align*}x=-3\end{align*}
- \begin{align*}x=0\end{align*}
- \begin{align*}y=0\end{align*}
In 25 – 28, find the intercepts of each equation.
- \begin{align*}y=4x-5\end{align*}
- \begin{align*}5x+5y=20\end{align*}
- \begin{align*}x+y=7\end{align*}
- \begin{align*}8y-16x=48\end{align*}
In 29 – 34, graph each equation using its intercepts.
- \begin{align*}3x+7y=21\end{align*}
- \begin{align*}2y-5x=10\end{align*}
- \begin{align*}x-y=4\end{align*}
- \begin{align*}16x+8y=16\end{align*}
- \begin{align*}x+9y=18\end{align*}
- \begin{align*}7+y=\frac{1}{7} x\end{align*}
In 35 – 44, find the slope between the sets of points.
- (3, 20) and (19, 8)
- (12, 5) and (12, 0)
- \begin{align*}\left (\frac{-1}{2},5\right )\end{align*} and \begin{align*}\left (\frac{3}{2},3\right )\end{align*}
- (8, 3) and (12, 3)
- (14, 17) and (–14, –22)
- (1, 4) and (18, 6)
- (10, 6) and (10, –6)
- (–3, 2) and (19, 5)
- (13, 9) and (2, 9)
- (10, –1) and (–10, 6)
In 45 – 50, determine the rate of change.
- Charlene reads 150 pages in 3 hours.
- Benoit cuts 65 onions in 1.5 hours.
- Brad drives 215 miles in 3.9 hours.
- Reece completes 65 jumping jacks in one minute.
- Harriet is charged $48.60 for 2,430 text messages.
- Samuel can eat 65 hotdogs in 22 minutes.
In 51 – 55, identify the slope and the \begin{align*}y-\end{align*}intercept of each equation.
- \begin{align*}x+y=3\end{align*}
- \begin{align*}\frac{1}{3} x=7+y\end{align*}
- \begin{align*}y=\frac{2}{5} x+3\end{align*}
- \begin{align*}x=4\end{align*}
- \begin{align*}y=\frac{1}{4}\end{align*}
In 56 – 60, graph each equation.
- \begin{align*}y=\frac{5}{6} x-1\end{align*}
- \begin{align*}y=x\end{align*}
- \begin{align*}y=-2x+2\end{align*}
- \begin{align*}y=-\frac{3}{8} x+5\end{align*}
- \begin{align*}y=-x+4\end{align*}
In 61 – 63, decide whether the given lines are parallel.
- \begin{align*}3x+6y=8\end{align*} and \begin{align*}y=2x-8\end{align*}
- \begin{align*}y=x+7\end{align*} and \begin{align*}y=-7-x\end{align*}
- \begin{align*}2x+4y=16\end{align*} and \begin{align*}y=\frac{-1}{2} x+6\end{align*}
In 64 – 70, evaluate the function for the indicated value.
- \begin{align*}g(n)= -2|n - 3|;\end{align*} Find \begin{align*}g(7)\end{align*}.
- \begin{align*}h(a)=a^2-4a\end{align*}; Find \begin{align*}h(8)\end{align*}.
- \begin{align*}p(t)=3t+1\end{align*}; Find \begin{align*}p\left (\frac{1}{6}\right )\end{align*}.
- \begin{align*}g(x)=4|x|\end{align*}; Find \begin{align*}g(-3)\end{align*}.
- \begin{align*}h(n)=\frac{1}{3} n-4\end{align*}; Find \begin{align*}h(24)\end{align*}.
- \begin{align*}f(x)=\frac{x+8}{6}\end{align*}; Find \begin{align*}f(20)\end{align*}.
- \begin{align*}r(c)=0.06(c)+c\end{align*}; Find \begin{align*}r(26.99)\end{align*}.
- The distance traveled by a semi-truck varies directly with the number of hours it has been traveling. If the truck went 168 miles in 4 hours, how many miles will it go in 7 hours?
- The function for converting Fahrenheit to Celsius is given by \begin{align*}C(F)=\frac{F-32}{1.8}\end{align*}. What is the Celsius equivalent to \begin{align*}84^\circ F\end{align*}?
- Sheldon started with 32 cookies and is baking more at a rate of 12 cookies/30 minutes. After how many hours will Sheldon have 176 cookies?
- Mixture \begin{align*}A\end{align*} has a 12% concentration of acid. Mixture \begin{align*}B\end{align*} has an 8% concentration of acid. How much of each mixture do you need to obtain a 60-ounce solution with 12 ounces of acid?
- The amount of chlorine needed to treat a pool varies directly with its size. If a 5,000-gallon pool needs 5 units of chlorine, how much is needed for a 7,500-gallon pool?
- The temperature (in Fahrenheit) outside can be predicted by crickets using the rule, “Count the number of cricket chirps in 15 seconds and add 40.”
- (i) Convert this expression to a function. Call it
- \begin{align*}T(c)\end{align*}
- , where
- \begin{align*}T=\end{align*}
- temperature
- and
- \begin{align*}c=\end{align*}
- number of chirps in 15 seconds.
- (ii) Graph this function. (iii) How many chirps would you expect to hear in 15 seconds if the temperature were
- \begin{align*}67^\circ F\end{align*}
- ? (iv) What does the
- \begin{align*}y-\end{align*}
- intercept mean? (v) Are there values for which this graph would not predict well? Why?
Graphing Linear Equations and Functions Test
Give the location of the following ordered pairs using the graph below.
- \begin{align*}A\end{align*}
- \begin{align*}B\end{align*}
- \begin{align*}C\end{align*}
- Graph \begin{align*}y=\frac{7}{3} x-4\end{align*}.
- Find the slope between (–3, 5) and (–1.25, –2.25).
- Find the intercepts of \begin{align*}6x+9y=54\end{align*}.
- In 2004, the high school graduation rate in the state of New Jersey was 86.3%. In 2008, the high school graduation rate in New Jersey was 84.6%. Determine the average rate of change. Use this information to make a conclusion regarding the graduation rate in New Jersey (source: http://nces.ed.gov/pubs2010/2010341.pdf).
- Identify the slope and \begin{align*}y-\end{align*}intercept of \begin{align*}4x+7y=28\end{align*}.
- Identify the slope and \begin{align*}y-\end{align*}intercept of \begin{align*}y=\frac{-3}{5} x-8\end{align*}.
- Graph \begin{align*}y=9\end{align*}.
- Graph the line containing (3, 5) and (3, –7). What type of line have you created?
- The number of cups of milk is directly proportional to the number of quarts. If 26 quarts yields 104 cups of milk, how many cups of milk is 2.75 quarts?
- Graph the direct variation situation using the table below:
\begin{align*}&0 && 1 && 2 && 3 && 4 && 5 && 6 && 7\\ & 0 && 2.25 && 4.5 && 6.75 && 9 && 11.25 && 13.5 && 15.75\end{align*}
- \begin{align*}h\end{align*} varies directly as \begin{align*}m\end{align*}, and when \begin{align*}m=4,h=27\end{align*}. Find \begin{align*}h\end{align*} when \begin{align*}m=-5.5\end{align*}.
- \begin{align*}h(n)=\frac{1}{2} |6-n|+11\end{align*}; Find \begin{align*}h(25)\end{align*}.
- Are these lines parallel? \begin{align*}y=-2x+1\end{align*} and \begin{align*}y=2x-1\end{align*}
- Mixture \begin{align*}A\end{align*} has a 2% hydrogen solution and mixture \begin{align*}B\end{align*} has a 1.5% hydrogen solution. How much of mixture \begin{align*}B\end{align*} needs to be added to 6 ounces of mixture \begin{align*}A\end{align*} to obtain a value of 0.51?
Texas Instruments Resources
In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9614.