Linear Function: Fitting Equations to Tables with Values (-1,10), (0,8), (1,6)

Q: Why is the slope negative when the table shows decreasing y-values?

A negative slope means the line goes downward from left to right. As x increases from -1 to 0 to 1, y decreases from 10 to 8 to 6. This creates a downward trend, which always gives a negative slope!

Q: How do I know which point to use for finding the y-intercept?

The y-intercept occurs when x = 0. In this table, when x = 0, y = 8, so the y-intercept is 8. If x = 0 isn't in your table, use any point with the slope in $ y = mx + b $ to solve for b.

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

Yes! Since it's a linear function, any two points will give you the same slope. Try using (-1, 10) and (1, 6): $ m = \frac{6-10}{1-(-1)} = \frac{-4}{2} = -2 $. Same result!

Q: How do I check if my equation is correct?

Substitute all three points into your equation. For $ y = -2x + 8 $: When x = -1, y = -2(-1) + 8 = 10 ✓. When x = 0, y = -2(0) + 8 = 8 ✓. When x = 1, y = -2(1) + 8 = 6 ✓

Q: What if I get a different equation form?

Linear equations can be written in different forms! $ y = -2x + 8 $ is slope-intercept form. You might also see $ 2x + y = 8 $ (standard form). Both represent the same line!

Linear Functions with Negative Slopes

Given the two tables of values x and and.

These tables represent a linear function. Fit an equation of a linear function to each one.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the algebraic representation for the given table

00:03 Let's take 2 given points

00:11 Use the formula to find the slope using 2 points on the graph

00:17 Substitute appropriate values according to the given data, and solve to find the slope

00:29 This is the graph's slope

00:37 Use the formula to represent a linear function

00:41 Substitute the point and solve for the unknown B

00:52 This is the Y-axis intersection point (unknown B)

00:58 Accordingly substitute the slope and intersection point to find the function

01:07 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the two tables of values x and and.

These tables represent a linear function. Fit an equation of a linear function to each one.

Step-by-step solution

To solve for the linear function, we need to follow these structured steps:

Step 1: Calculate the slope ( $m$ ).

Use the points $(-1, 10)$ and $(0, 8)$ . The slope $m = \frac{8 - 10}{0 - (-1)} = \frac{-2}{1} = -2$ .

Step 2: Determine the y-intercept ( $b$ ).

Use the point-slope form with the point $(0, 8)$ (since $x = 0$ is the y-intercept directly). Thus, $b = 8$ .

Step 3: Write the equation of the line.

The equation fitting the table values is $y = -2x + 8$ .

Therefore, the equation of the linear function is $y = -2x + 8$ .

Final Answer

$y=-2x+8$

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use any two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Technique: Find slope first: $m = \frac{8-10}{0-(-1)} = \frac{-2}{1} = -2$
Check: Substitute all points into final equation: $y = -2x + 8$ ✓

Common Mistakes

Avoid these frequent errors

Mixing up coordinates when calculating slope
Don't switch x and y values or use points in wrong order = wrong slope sign! This gives you a positive slope when it should be negative. Always write points clearly as (x₁, y₁) and (x₂, y₂) and subtract consistently.

Practice Quiz

Test your knowledge with interactive questions

For the function in front of you, the slope is?

FAQ

Everything you need to know about this question

Why is the slope negative when the table shows decreasing y-values?

A negative slope means the line goes downward from left to right. As x increases from -1 to 0 to 1, y decreases from 10 to 8 to 6. This creates a downward trend, which always gives a negative slope!

How do I know which point to use for finding the y-intercept?

The y-intercept occurs when x = 0. In this table, when x = 0, y = 8, so the y-intercept is 8. If x = 0 isn't in your table, use any point with the slope in $y = mx + b$ to solve for b.

Can I use any two points to find the slope?

Yes! Since it's a linear function, any two points will give you the same slope. Try using (-1, 10) and (1, 6): $m = \frac{6-10}{1-(-1)} = \frac{-4}{2} = -2$ . Same result!

How do I check if my equation is correct?

Substitute all three points into your equation. For $y = -2x + 8$ : When x = -1, y = -2(-1) + 8 = 10 ✓. When x = 0, y = -2(0) + 8 = 8 ✓. When x = 1, y = -2(1) + 8 = 6 ✓

What if I get a different equation form?

Linear equations can be written in different forms! $y = -2x + 8$ is slope-intercept form. You might also see $2x + y = 8$ (standard form). Both represent the same line!

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