Linear Function Graph Through Points (4,7) and (7,2)

Q: What does a negative slope actually mean?

A negative slope means the line goes downward from left to right. As x-values increase, y-values decrease. Think of walking down a hill - that's a negative slope!

Slope Analysis with Two Given Points

The graph of the linear function passes through the points $B(4,7),A(7,2)$

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Step-by-step video solution

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00:08 First, let's determine what type of slope we have.

00:12 Next, we find the slope using two points.

00:23 Use the formula. Remember, slope equals the change in Y over the change in X.

00:31 Now, substitute the values from your points into the formula to calculate the slope.

00:46 Since the slope is negative, the function is decreasing.

00:54 And that's how we solve this problem!

Step-by-step written solution

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Understand the problem

The graph of the linear function passes through the points $B(4,7),A(7,2)$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Calculate the slope using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Given points are $B(4,7)$ and $A(7,2)$ . Therefore, $x_1 = 4$ , $y_1 = 7$ , $x_2 = 7$ , $y_2 = 2$ .
Step 2: Substitute the values into the formula:
$m = \frac{2 - 7}{7 - 4} = \frac{-5}{3} = -\frac{5}{3}$ .
Step 3: Determine the nature of the function based on the slope:
Since $m = -\frac{5}{3}$ is negative, the function is a decreasing function.

Therefore, based on the slope being negative, the function represented by the line passing through these points is a Decreasing function.

Final Answer

Decreasing function

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use m = (y₂ - y₁)/(x₂ - x₁) for any two points
Calculation: m = (2 - 7)/(7 - 4) = -5/3 shows negative slope
Check: Negative slope means y decreases as x increases ✓

Common Mistakes

Avoid these frequent errors

Confusing coordinate order when calculating slope
Don't mix up which point is (x₁, y₁) and which is (x₂, y₂) = wrong slope sign! This changes whether your function increases or decreases. Always label your points clearly and subtract coordinates in the same order: (y₂ - y₁) and (x₂ - x₁).

Practice Quiz

Test your knowledge with interactive questions

What is the solution to the following inequality?

$$ 10x-4\leq-3x-8 $$

FAQ

Everything you need to know about this question

How do I know which point to call point 1 and point 2?

It doesn't matter which point you choose as point 1! The slope will be the same either way. Just make sure you're consistent - if B(4,7) is point 1, then A(7,2) must be point 2.

What does a negative slope actually mean?

A negative slope means the line goes downward from left to right. As x-values increase, y-values decrease. Think of walking down a hill - that's a negative slope!

Could this function be constant instead of decreasing?

No! A constant function has slope = 0, meaning all y-values are the same. Since our points have different y-values (7 and 2), the function must be increasing or decreasing.

What if I got a positive slope instead?

Check your arithmetic! With points (4,7) and (7,2), you should get $m = \frac{2-7}{7-4} = \frac{-5}{3}$ . A common error is forgetting the negative sign when subtracting.

How can I visualize this without graphing?

Compare the y-coordinates: point B has y = 7, point A has y = 2. Since we move from a higher y-value to a lower y-value as x increases, the function is decreasing!

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