Linear Functions
Linear Functions
See one relationship four ways: a situation, a table, a graph, and an equation. Then learn how input, output, rate of change, and intercepts fit together.
1. What is a function?
A function is a rule that assigns exactly one output to every allowed input.
Once the input is fixed, a function cannot produce two different outputs.
Function notation
If
then the input is \(x\), and the output is \(f(x)\).
The key test
Ask:
Can one input lead to two different outputs?
If yes, it is not a function.
2. A function in the real world: the taxi model
Suppose a taxi charges a $4 starting fee plus $2.25 per mile. What changes? What stays fixed?
What changes?
The cost changes as the number of miles changes.
That makes miles the input.
What stays fixed?
The $4 starting fee is charged even when the trip is 0 miles.
That is the initial value.
Here, \(x\) is the number of miles and \(C(x)\) is the total cost.
miles traveled
total taxi cost
3. The coordinate plane: where functions live visually
A graph gives us a picture of how the output changes as the input changes. The horizontal axis is the x-axis; the vertical axis is the y-axis.
x-intercept
The point where a graph crosses the x-axis. Because every point on the x-axis has \(y=0\), the x-intercept has the form \((a,0)\).
y-intercept
The point where a graph crosses the y-axis. Because every point on the y-axis has \(x=0\), the y-intercept has the form \((0,b)\).
4. What does the slope of a line mean?
Slope measures the rate at which \(y\) changes when \(x\) changes.
On a graph, slope can be read as rise over run.
Positive slope
\(m>0\). The line rises from left to right.
Negative slope
\(m<0\). The line falls from left to right.
Zero slope
\(m=0\). The line is horizontal.
5. Three important line shapes
6. Horizontal and vertical lines
These two special equations are worth recognizing immediately.
Horizontal line
The output is always the same. No matter what \(x\) is, the value of \(y\) stays at \(a\). Its slope is 0.
Vertical line
The input is fixed at \(b\). The line is vertical, so it does not represent \(y\) as a function of \(x\).
7. Function or not a function?
Let’s make the definition visual. In a function, one input can have only one output.
Each input has exactly one output.
The input 1 produces both 5 and 7. That breaks the function rule.
8. One function, four representations
The SAT often asks you to connect different representations of the same relationship. College Board specifically identifies connections among input/output pairs, tables, graphs, and algebraic representations as part of the linear-functions skill.
A gym charges a $20 registration fee plus $12 per month.
| \(m\) | 0 | 1 | 2 |
|---|---|---|---|
| \(G(m)\) | 20 | 32 | 44 |
9. Reading slope and intercept from a linear function
\(m\): slope
How much \(y\) changes for each 1-unit increase in \(x\).
\(b\): y-intercept
The output when \(x=0\). The graph crosses the y-axis at \((0,b)\).
Here the line has slope 3 and y-intercept 5. Starting at the y-intercept, move right 1 and up 3 to reach another point on the line.
10. Finding a linear rule from two input/output pairs
Suppose a function passes through \((2,11)\) and \((6,23)\).
\(11=6+b\)
\(b=5\)
11. Work forward — and work backward
Given the input
If \(f(x)=4x-3\), find \(f(7)\).
Given the output
If \(f(x)=4x-3\) and \(f(x)=25\), find \(x\).
12. Original SAT-style practice
These questions are newly written for SATMath800. They deliberately use contexts and numbers that are different from the worked examples above.
A function is defined by \(p(x)=5x-7\). What is the value of \(p(9)\)?
- A) 38
- B) 42
- C) 45
- D) 52
A linear function has values \(q(3)=8\) and \(q(7)=20\). What is the rate of change of \(q\)?
- A) 2
- B) 3
- C) 4
- D) 6
A delivery company charges a fixed fee plus $4 for each mile. The equation \(D(m)=4m+9\) gives the total charge in dollars for \(m\) miles. What does 9 represent?
- A) The number of miles traveled
- B) The charge per mile
- C) The total charge for 9 miles
- D) The fixed fee
Which equation represents a horizontal line that passes through \((0,-6)\)?
- A) \(x=-6\)
- B) \(y=6\)
- C) \(y=-6\)
- D) \(y=x-6\)
Which equation represents a vertical line through \(x=4\)?
- A) \(y=4\)
- B) \(x=4\)
- C) \(y=x+4\)
- D) \(x=y+4\)
A line passes through \((-2,7)\) and \((4,-5)\). What is its slope?
- A) \(-3\)
- B) \(-2\)
- C) \(2\)
- D) \(3\)
Which relation is not a function of \(x\)?
- A) \(y=2x+1\)
- B) \(y=-4\)
- C) \(x=3\)
- D) \(y=\dfrac{x}{2}\)
A function satisfies \(r(4)=13\) and \(r(10)=31\). What is \(r(16)\)?
- A) 43
- B) 46
- C) 49
- D) 52
Temperature conversion is modeled by \(F=\dfrac{9}{5}C+32\). If the temperature is \(68^\circ\!F\), what is the temperature in degrees Celsius?
- A) 15
- B) 18
- C) 20
- D) 24
A line has slope \(0\) and passes through \((5,12)\). Which equation represents the line?
- A) \(x=12\)
- B) \(y=5\)
- C) \(y=12\)
- D) \(y=5x+12\)
13. Mastery check
Identify
Can you identify the input, output, slope, and intercept?
Connect
Can you move between a situation, table, graph, and equation?
Reason
Can you recognize whether a relation is a function and explain why?