Grade 8

Intermediate

10 min

Lesson 6 of 12

Comparing Functions

Not Started

Analyze and compare two different functions represented in different ways.

Imagine two delivery services: 'Turbo-Track' charges a flat $5 fee plus$ 2 per mile, while 'Swift-Ship' has no flat fee but charges $3 per mile. Which one is cheaper for a 10-mile trip? Without comparing their functions, you might end up paying double!

Learning Objectives

1.By the end of this lesson, you will be able to calculate and compare rates of change from tables, graphs, and equations.
2.You'll understand how to identify and compare initial values ( $y$ -intercepts) across different function representations.
3.You'll be able to determine which function grows faster or starts higher in real-world scenarios.

The Speed of Change: Rate of Change

In any linear relationship, the rate of change (also called the slope) tells us how much the

y

-value changes for every 1 unit increase in

x

. When comparing functions, the one with the higher rate of change is 'steeper' on a graph or 'faster' in a real-world context. To find the rate of change from a table, use the formula:

\begin{gathered}m = \\frac{y_2 - y_1}{x_2 - x_1}\end{gathered}

On a graph, you can simply count the rise over run. If you have an equation in the form

y = mx + b

, the rate of change is simply the coefficient

m

Compare the rate of change for these two functions: 1. Function A: $y = 4x + 10$ 2. Function B: A table where $x=1, y=5$ and $x=2, y=8$ .

Step 1: Identify the slope of Function A. In

y = 4x + 10

m = 4

. Step 2: Calculate the slope of Function B using the table values:

\begin{gathered}m = \\frac{8 - 5}{2 - 1} = \\frac{3}{1} = 3\end{gathered}

Step 3: Compare. Since

4 > 3

, Function A has a greater rate of change.

Quick Check

If a function's table shows that $y$ increases by $12$ every time $x$ increases by $3$ , what is its rate of change?

Divide the change in

y

by the change in

x

Answer

The rate of change is $4$ .

The Starting Line: Initial Value

The initial value is the $y$ -value when $x = 0$ . On a graph, this is the y-intercept, the exact point where the line crosses the vertical axis. In a table, look for the row where $x = 0$ . In the equation $y = mx + b$ , the initial value is the constant $b$ . Comparing initial values is crucial for understanding 'starting costs' or 'starting positions.' Even if a function has a slower rate of change, it might still be 'ahead' for a while if its initial value is much higher.

Two hikers are walking. - Hiker A starts at the 5-mile marker and walks 2 mph. - Hiker B is represented by the equation $y = 3x + 2$ , where $y$ is the mile marker and $x$ is hours.

Step 1: Find the initial value for Hiker A. They start at the 5-mile marker, so $b = 5$ . Step 2: Find the initial value for Hiker B from the equation $y = 3x + 2$ . Here, $b = 2$ . Step 3: Compare. Hiker A has a greater initial value ( $5 > 2$ ), meaning they started further along the trail.

Quick Check

Where do you look on a graph to find the initial value of a function?

Answer

You look at the y-intercept, which is where the line crosses the y-axis (where $x=0$ ).

The Full Comparison

To fully compare two functions, you must look at both the rate of change and the initial value. A function might start lower (smaller $b$ ) but eventually overtake another function if it has a higher rate of change (larger $m$ ). This 'break-even' point is where the two functions are equal. When solving word problems, always identify your $m$ and $b$ for both functions first, then compare them based on what the question asks (e.g., 'Which is cheaper?' or 'Which is faster?').

Gym A charges a $50 sign-up fee and$ 10 per month. Gym B is shown on a graph that passes through $(0, 20)$ and $(2, 60)$ . Which gym is cheaper after 6 months?

1. Gym A: $y = 10x + 50$ . At 6 months: $y = 10(6) + 50 = 110$ . 2. Gym B: Initial value is $20$ . Rate of change is $\\frac{60-20}{2-0} = 20$ . Equation: $y = 20x + 20$ . At 6 months: $y = 20(6) + 20 = 140$ . 3. Conclusion: Gym A is cheaper for a 6-month membership ( $110 < 140$ ).

Key Takeaways

1The rate of change ( $m$ ) determines how steep the line is or how fast the value increases.
2The initial value ( $b$ ) is the starting point where $x = 0$ .
3To compare functions in different formats, convert them to the same format (like $y = mx + b$ ) or calculate $m$ and $b$ for both.
4A higher rate of change means the function will eventually have higher $y$ -values, regardless of the starting point.

Quick Check

Multiple Choice

Function 1 is $y = 6x + 2$ . Function 2 is a table with points $(0, 5)$ and $(1, 10)$ . Which has a greater rate of change?

Multiple Choice

Which function has the greatest initial value?

True/False

If Function A has a higher rate of change than Function B, it will always have a higher $y$ -value for any $x > 0$ .

What's Next?

Review Tomorrow

In 24 hours, try to explain to a friend how to find the 'starting cost' and 'monthly rate' of a phone plan using only a graph.

Practice Activity

Look at a recent utility bill or a subscription service. Can you identify the flat fee (initial value) and the usage rate (rate of change)?

Browse

Browse

Comparing Functions

Learning Objectives

The Speed of Change: Rate of Change

The Starting Line: Initial Value

The Full Comparison

Key Takeaways

Quick Check

What's Next?

Comparing Functions

Learning Objectives

The Speed of Change: Rate of Change

The Starting Line: Initial Value

The Full Comparison

Key Takeaways

Quick Check

What's Next?