Breaking Down Linear Function Problems on the SAT

Linear function questions are some of the trickiest algebra problems on the SAT math section. They describe real-world scenarios modeled by linear equations, requiring strong comprehension skills to unpack the context.

Students must know how to define variables to represent quantities and set up the function that fits the relationship described. They need to methodically manipulate the function to analyze it - for example, evaluating it for a given input or identifying the meaning of constants.

The key to tackling these difficult problems is first understanding the context, then breaking down the steps - define variables, set up the function, manipulate and evaluate it. By improving comprehension of word problems and methodically following the analytical steps, these problems become a lot more manageable.

Now, let's take a look at some examples

Example 1:

The number of printers sold by a company dropped from 8,000 in 2010 to 3,800 in 2017. Assuming printer sales decreased at a constant rate, which linear function p best models the number of printers sold t years after 2010?

Okay, let's break this down:

1) Write the key points:

In 2010, 8,000 printers were sold

In 2017, 7 years later, 3,800 printers were sold

Recognize this is a linear relationship - the sales are decreasing at a steady rate over time. To write a linear function, we need the slope and the y-intercept.

2) To find the slope, use the change in y over the change in x:

Change in printers sold: 8,000 - 3,800 = 4,200

Change in years: 2017 - 2010 = 7

Slope = Change in y / Change in x = 4,200/7 = -600

3) To find the y-intercept, plug in the initial point:

In 2010, t = 0, and 8,000 printers were sold

So when t = 0, p = 8,000

The linear function is: $p = -600t + 8,000$

Therefore, the function $p = -600t + 8,000$ best models the printer sales over time.

By recognizing the relationship is linear, finding the slope and y-intercept, and writing the equation, we can methodically break down linear function word problems on the SAT. With practice, you'll become very comfortable solving these types of questions.

Example 2:

One bag of mulch covers 12 square feet of garden space. A flower bed has a total area of a square feet. Which equation represents the total number of bags, B, needed to cover the flower bed with 2 layers of mulch?

A) B * 12 = a
B) B * 24 = a
C) B + 12 = a
D) a / 12 = B

Okay, here is how to break this down:

1) Write the key info:

1 bag covers 12 sq ft

The flower bed area is a sq ft

We need 2 layers of mulch

2) To find the bags needed:

The bed needs to be covered twice

So if 1 bag covers 12 sq ft, 2 bags cover 2 * 12 = 24 sq ft

3) Set up an equation:

Let B = number of bags

Each bag covers 24 sq ft (2 layers)

The total area is a sq ft

Therefore, the equation representing the total number of bags needed is $B * 24 = a$

By methodically writing the key points, determining the relationship, and setting up an equation, we can tackle linear relationship word problems on the SAT. With practice, these types of questions will feel very manageable.

Example 3:

Evaluating a function for a given input is a common problem seen on the SAT math section. Let's look at an example:

For the function $g$, if $g(2x) = 3x - 4$ for all $x$, what is $g(8)$?

To break this down:

1) Understand the given information

We are given a function g that relates $g(2x)$ to $3x - 4$
This tells us that whatever is input into the function $g$, the input gets multiplied by 3 and subtracted by 4.

2) Find $x$

This means we need to take the input 8 and substitute it into the function $g$. To substitute 8 into $g$, we set $8 = 2x$, so $x = 4$.

3) Solve for $g(8)$

Now that we know the value of $x$, we can take it and multiply it by 3 and then subtract by 4:
$3(4) - 4 = 8$. So, $g(8) = 8$

Key Takeaways

The key points to understand when solving linear functions are:

• Understand how the function transforms the input
• Substitute the given input value by applying the function steps
• Simplify the result

As you can see, with a methodical approach and an understanding of the basic concepts, even the most challenging SAT algebra word problems become solvable. The key is to stay persistent, practice frequently, and always look for ways to break down the problem step-by-step.