Linear, Exponential, and Quadratic Functions

What is a function?

A function is a rule that relates inputs (x) to outputs (y). 

It assigns exactly one output for each input, defining a relationship between the two.

For example, hours worked can be the input (x), pay earned can be the output (y), and the pay rule is the function. The pay (dependent variable) depends on the hours put in (independent variable).

Functions can be represented with an equation, table, or graph. Its graph passes a vertical line test.

What is a constant addend?

A constant addend is the same amount added to the y-values each time the x-values increase by 1.

This creates a linear relationship between the inputs and outputs.

What is a constant multiplier?

A constant multiplier is the same number multiplied to the y-values each time the x-values increase by 1.

This creates an exponential relationship between inputs and outputs.

What is slope?

The slope of a line measures steepness and rate of change.

It is found by change in y divided by change in x (rise over run).

It is represented by m in y = mx + b.

What is y-intercept?

The y-intercept is the point where a graph crosses the y-axis.

It shows the value of y when x is zero.

For a linear equation, the y-intercept is b.

For an exponential equation, the y-intercept is a.

It is often the starting or initial value.

What is a linear function?

A linear function has a highest exponent of 1 on the variable.

It has a constant addend.

It graphs a straight line.

It is represented by the equation y = mx + b, where m is the slope and b is the y-intercept.

It shows a constant rate of change between the x and y variables.

Its parent function equation is y=x.

What is an exponential function?

An exponential function has the x variable in the exponent.

It has a constant multiplier.

It graphs a curve.
It is represented by the equation $y=a\left(b\right)^{x}$ , where a is the initial value, b is the base (a constant multiplier greater than 0), and x is the exponent.

It typically describes a rapid growth or decay process, with the rate of change increasing or decreasing exponentially.

Its parent function equation is $y=2^{x}$

What is percent increase?

Percent increase measures how much a value has grown compared to its original amount, expressed as a percentage.

It is found by subtracting the original value from the new value, dividing by the original value, and multiplying by 100.

In exponential growth, it can also be found by subtracting 1 from the multiplier and multiplying by 100. (b-1)x100=percent increase

Percent increase is often used to describe the growth rate of a quantity over time, especially in financial contexts or population studies.

What is a percent decrease?

Percent decrease is a measure of how much a value has decreased relative to its original amount, expressed as a percentage.

It is calculated by taking the difference between the original value and the new value, dividing that difference by the original value, and then multiplying by 100 to get the percentage.

In exponential decay, it can be found by subtracting the multiplier from 1 and multiplying by 100. (1-b)x100=percent decrease.

What is a ratio?

A ratio is a comparison of two quantities using division. It shows the relationship between the two amounts.

What is a quadratic function?

A quadratic function has a highest exponent of 2 on the variable.
It has a second difference.

It graphs a parabola.

It can be written as $y=ax^2+bx+c$ and graphs as a parabola (u-shape).

It may rise then fall or fall then rise.
Its parent function equation is $y=x^2$

What is second difference?

The second difference is the difference of the differences between consecutive y-values in a sequence each time the x-values increase by 1.

This creates an quadratic relationship between inputs and outputs.

What does difference mean?

The difference refers to the result of subtracting one quantity from another.