Review: Linear functions from a table

What is a linear function?

A linear function is a function that creates a straight line when graphed. It has the form $\displaystyle y = mx + b$, where $\displaystyle m$ is the slope and $\displaystyle b$ is the y-intercept.

Difference between slope and y-intercept.

The slope ($\displaystyle m$) indicates the steepness of the line, while the y-intercept ($\displaystyle b$) is the value of $\displaystyle y$ when $\displaystyle x = 0$.

True or false: All linear functions have a constant slope.

True, because linear functions maintain the same rate of change throughout.

Fill in the blank: A table showing input-output pairs for a linear function is called a _____.

function table

How do you find the slope from a table?

Calculate the change in $\displaystyle y$ over the change in $\displaystyle x$: $\displaystyle m = \\frac{\Delta y}{\Delta x}$.

What do you look for in a linear function table?

You look for a consistent rate of change in $\displaystyle y$ as $\displaystyle x$ increases.

Example of a real-world linear function: Rent.

If rent increases by $\displaystyle 200 each year, it can be modeled as$y = 200x + b$\displaystyle , where$b$ is the initial rent.

Question: How to determine if a table represents a linear function?

Check if the differences in $\displaystyle y$ values are constant for equal changes in $\displaystyle x$ values.

Compare linear and non-linear functions.

Linear functions graph as straight lines; non-linear functions graph as curves or other shapes.

True or false: A linear function can have a negative slope.

True, because a negative slope indicates a decrease in $\displaystyle y$ as $\displaystyle x$ increases.

Real-life scenario: Trip cost.

Cost can be expressed as $\displaystyle C = 0.50d + 5$, where $\displaystyle d$ is distance in miles and $\displaystyle 5$ is the base fare.

What does a horizontal line indicate?

A horizontal line indicates a slope of $\displaystyle 0$, meaning $\displaystyle y$ does not change as $\displaystyle x$ changes.

Quiz: What is the slope in the table below? | x: 1, 2, 3 | y: 3, 6, 9.

Slope $\displaystyle m = 3$ since $\displaystyle \Delta y / \Delta x = 3$.

Fill in the blank: In a linear equation, $\displaystyle y$ changes by ____ for every change in $\displaystyle x$.

the slope value

Example of a function table.

| x | y | | 1 | 2 | 3 | | 2 | 4 | 6 | shows linear growth.

Question: What does the y-intercept represent?

The y-intercept represents the value of $\displaystyle y$ when $\displaystyle x$ is zero.

True or false: A linear function cannot cross the x-axis.

False, because linear functions can cross the x-axis when $\displaystyle y = 0$.

Comparison of increasing and decreasing linear functions.

Increasing functions have a positive slope; decreasing functions have a negative slope.

Example of a paycheck model.

Paycheck can be modeled as $\displaystyle P = 15h$, where $\displaystyle h$ is hours worked at $15 per hour.