Identifying a Linear Function from an Equation

Is $y = 3x - 2$ a linear function?

Check the form: Compare to $y = mx + b$ = It matches the pattern

Identify the slope: The coefficient of $x$ is $3$ = $m = 3$

Identify the y-intercept: The constant term is $-2$ = $b = -2$

Verify linearity: No exponents on $x$, no $x$ in denominator = It is linear

Answer: Yes, $y = 3x - 2$ is a linear function with slope $3$ and y-intercept $-2$.

Creating a Table of Values

Create a table of values for $y = 2x + 1$ using $x = -1, 0, 1, 2$.

Substitute $x = -1$: $y = 2(-1) + 1 = -2 + 1$ = $y = -1$

Substitute $x = 0$: $y = 2(0) + 1 = 0 + 1$ = $y = 1$

Substitute $x = 1$: $y = 2(1) + 1 = 2 + 1$ = $y = 3$

Substitute $x = 2$: $y = 2(2) + 1 = 4 + 1$ = $y = 5$

Verify constant change: Each time $x$ increases by $1$, $y$ increases by $2$ = Rate of change is constant

Answer: Table: $(-1, -1)$, $(0, 1)$, $(1, 3)$, $(2, 5)$. Notice the constant rate of change of $2$.

Identifying Non-Linear Functions

Which of these is NOT a linear function? a) $y = 5x$ b) $y = x^2 + 1$ c) $y = -4x + 7$

Check $y = 5x$: Can be written as $y = 5x + 0$ = Linear (m=5, b=0)

Check $y = x^2 + 1$: Has $x^2$ (exponent of 2) = NOT linear - it's quadratic

Check $y = -4x + 7$: Matches $y = mx + b$ form = Linear (m=-4, b=7)

Answer: $y = x^2 + 1$ is NOT linear because the variable $x$ is squared. Linear functions only have $x$ to the first power.

Real-World Application

A taxi charges 3 dollars plus 2 dollars per mile. Write the linear function and find the cost for a 5-mile ride.

Identify the parts: Fixed cost: 3 dollars, Rate: 2 dollars per mile = b = 3, m = 2

Write the function: Let $x$ = miles, $y$ = total cost = $y = 2x + 3$

Substitute $x = 5$: $y = 2(5) + 3 = 10 + 3$ = $y = 13$

Interpret the answer: The total cost is 13 dollars = 13 dollars for 5 miles

Answer: The function is $y = 2x + 3$. A 5-mile ride costs 13 dollars.

Mistake: Thinking $y = x^2$ is linear because it has an $x$

Why: The exponent matters! $x^2$ means $x$ multiplied by itself, which creates a curve, not a straight line.

Correct: Linear functions have $x$ to the first power only. Check that there's no $x^2$, $x^3$, $\sqrt{x}$, or $\frac{1}{x}$.

Mistake: Confusing slope and y-intercept in $y = mx + b$

Why: It's easy to mix them up. The slope ($m$) is the coefficient of $x$, while $b$ is the constant term.

Correct: In $y = 3x + 5$, the slope is $3$ (multiplies $x$) and the y-intercept is $5$ (stands alone).

Mistake: Thinking $y = 5$ is not a linear function

Why: This is actually $y = 0x + 5$, a horizontal line with slope $0$.

Correct: $y = 5$ is linear - it's a special case with zero slope. The graph is a horizontal line at $y = 5$.

Cell Phone Plans

Many phone plans have a base fee plus a per-usage charge, creating a linear function.

A plan costs 25 dollars monthly plus 10 cents per minute over the limit. If $x$ is extra minutes, then $y = 0.10x + 25$.

Distance and Speed

When traveling at a constant speed, distance is a linear function of time.

Driving at 55 mph: $d = 55t$, where $d$ is distance in miles and $t$ is time in hours.

Earnings and Hours Worked

Hourly wages create a linear relationship between hours worked and money earned.

Earning 12 dollars per hour: $y = 12x$, where $x$ is hours and $y$ is earnings in dollars.

Q: What makes a function linear vs non-linear?

A: A function is linear if its graph is a straight line. Mathematically, this means the equation can be written as $y = mx + b$ with no exponents on variables, no variables in denominators, and no variables under radicals.

Q: Can the slope be zero or negative?

A: Yes! A slope of 0 gives a horizontal line ($y = b$). A negative slope means the line goes down from left to right (like a hill you ski down).

Q: What's the difference between a linear function and a linear equation?

A: A linear equation is any equation whose graph is a line. A linear function is specifically written with $y$ alone on one side: $y = mx + b$. Every linear function is a linear equation, but not every linear equation is written as a function.