Linear Function Verification: Is the Table of X = [-1, 2, 6] and Y = [1, 2, 3] Consistent?

Q: Why can't I just check if the y-values increase by the same amount?

Because linear functions depend on rate of change, not just change in y-values! The x-values also change by different amounts, so you need the slope formula to find the true rate.

Q: What if I get the same slope for all pairs?

Then congratulations - you have a linear function! Equal slopes mean constant rate of change, which is exactly what defines linearity.

Q: Do I need to check every single pair of points?

Yes, check all consecutive pairs! Even if the first two pairs have equal slopes, the third pair might be different. All slopes must be identical for the function to be linear.

Q: Can I use any two points to find the slope?

For verification, use consecutive points in order. This ensures you're checking the rate of change throughout the entire function, not just between random points.

Q: What does it mean if the slopes are different?

Different slopes mean the function is not linear! The rate of change varies, which could indicate a quadratic, exponential, or other type of function.

Linear Function Verification with Slope Calculations

Determine whether the following table represents a linear function

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Does the following table represent a function?

00:04 The function has a constant slope, constant difference between Y values

00:07 Let's look at the differences between Y values and try to find the slope

00:11 The differences are equal, the slope is constant, therefore it's a function

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Determine whether the following table represents a linear function

Step-by-step solution

To determine if the table represents a linear function, we need to check if the slope between each consecutive pair of points is constant.
Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ , we calculate:

Between points $(-1, 1)$ and $(2, 2)$ :
$m = \frac{2 - 1}{2 - (-1)} = \frac{1}{3}$
Between points $(2, 2)$ and $(6, 3)$ :
$m = \frac{3 - 2}{6 - 2} = \frac{1}{4}$

Since the slopes are not equal ( $\frac{1}{3} \neq \frac{1}{4}$ ), the function is not linear.

Thus, the table does not represent a linear function.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Linear Definition: All consecutive points must have identical slopes
Slope Formula: Calculate $m = \frac{y_2 - y_1}{x_2 - x_1}$ between each pair
Check: Compare all slopes: $\frac{1}{3} \neq \frac{1}{4}$ means not linear ✓

Common Mistakes

Avoid these frequent errors

Only checking if points increase steadily
Don't assume steady increases mean linear = wrong conclusion! Just because y-values go 1, 2, 3 doesn't guarantee linearity. Always calculate actual slopes between consecutive points to verify constant rate of change.

Practice Quiz

Test your knowledge with interactive questions

Determine whether the given graph is a function?

FAQ

Everything you need to know about this question

Why can't I just check if the y-values increase by the same amount?

Because linear functions depend on rate of change, not just change in y-values! The x-values also change by different amounts, so you need the slope formula to find the true rate.

What if I get the same slope for all pairs?

Then congratulations - you have a linear function! Equal slopes mean constant rate of change, which is exactly what defines linearity.

Do I need to check every single pair of points?

Yes, check all consecutive pairs! Even if the first two pairs have equal slopes, the third pair might be different. All slopes must be identical for the function to be linear.

Can I use any two points to find the slope?

For verification, use consecutive points in order. This ensures you're checking the rate of change throughout the entire function, not just between random points.

What does it mean if the slopes are different?

Different slopes mean the function is not linear! The rate of change varies, which could indicate a quadratic, exponential, or other type of function.

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