Linear, Quadratic, and Exponential Models

Functions are a type of mathematical relationship between inputs and outputs. In this guide, you will learn more about the types of functions.

Tutor-style math help

Linear, Quadratic, and Exponential Models: what to notice and how to work it

Exponential skill

Exponential functions change by repeated multiplication. The base is the multiplier, and the starting value is usually the output when the exponent is zero.

What to notice first

Modeling questions are about choosing the correct type of change: constant addition, repeated multiplication, or a quadratic pattern.

Common student mistake

Do not decide from one pair of values. Check whether the change stays constant or the multiplier stays constant.

Key formulas and cues

\(y=a\cdot b^x\)

\(b>1\Rightarrow\text{growth}\)

\(0<b<1\Rightarrow\text{decay}\)

\(A=P(1+r)^t\)

A reliable path

Find the startLook for the value when x or time is 0.
Find the multiplierUse the base or percent change to identify b.
Check the shapeGrowth rises away from the asymptote; decay moves toward it.

Worked examples

Growth model

Example: 100 grows by a factor of 2 for 3 steps

Start at 100.
Multiply by 2 three times.
Compute 100 times 8.

Answer: \(800\)

Decay model

Example: 80 is cut in half twice

Half of 80 is 40.
Half of 40 is 20.
This is multiplying by 1/2 twice.

Answer: \(20\)

Open pop-up practice

Try one before moving on

Try: A value starts at 50 and triples twice. What is it?

Answer: \(450\). Compute \(50\cdot3^2\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

A step-by-step guide to linear, quadratic, and exponential models

Functions are a type of mathematical relationship between inputs and outputs. The linear, quadratic, and exponential functions are the \(3\) main types of functions.

Linear function:

A linear function is a type of function whose highest exponent is \(1\). The standard form of this function is \(y=mx + b\). The graph of the linear function shows a straight line in the coordinate plane.

Quadratic function:

A quadratic function is a polynomial function with one or more variables, where the highest power of the variable is two. Since the highest degree in a quadratic function is of the second degree, it is also called the polynomial of degree \(2\).

The quadratic function, when graphed, makes parabolas. The standard form of a quadratic function is \(f(x) = ax^2+bx + c\), where \(a, b,\) and \(c\) are real numbers with \(a≠0\).

Exponential function:

An exponential function is a function of the form \(f(x)= a^x\), where \(x\) is a variable and \(a\) is a constant, which is called the base of the function and must be greater than \(0\). Exponential function graphs have curves. This curve can be vertical at first and then grow horizontally, or it can be horizontal at first and then become more vertical.

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Linear, Quadratic, and Exponential Models