Linear Function Through Points (5½,10) and (½,5): Coordinate Analysis

Q: How do I handle mixed numbers like 5½ in coordinates?

Convert mixed numbers to decimals for easier calculation! $ 5\frac{1}{2} = 5.5 $ and $ \frac{1}{2} = 0.5 $. This makes the slope formula much simpler to work with.

Q: What's the difference between bottom-up and decreasing functions?

A bottom-up function means increasing (positive slope) - as x increases, y increases too. A decreasing function has negative slope - as x increases, y decreases. Don't confuse the terms!

Q: Why does positive slope = 1 make this a bottom-up function?

When slope is positive, the line goes up from left to right. Since m = 1 > 0, this is an increasing or "bottom-up" function. The line rises as you move right.

Q: Do I always need to simplify the slope fraction?

Yes! Always simplify your slope to lowest terms. In this problem, $ \frac{5}{5} = 1 $, which clearly shows the slope is exactly 1.

Q: Can a linear function be constant if it passes through different points?

No! A constant function has slope = 0, meaning all points have the same y-value. Since our points have different y-values (5 and 10), this cannot be constant.

Slope Analysis with Mixed Number Coordinates

The graph of the linear function passes through the points $B(5\frac{1}{2},10),A(\frac{1}{2},5)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the type of slope

00:04 Find the slope using 2 points

00:21 Use the formula to find the slope using 2 points

00:27 Substitute appropriate values according to the given data and solve to find the slope

00:49 The slope is positive, therefore the function is increasing

00:58 And this is the solution to the question

Step-by-step written solution

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Understand the problem

The graph of the linear function passes through the points $B(5\frac{1}{2},10),A(\frac{1}{2},5)$

Step-by-step solution

To determine the nature of the linear function, let's calculate the slope of the line passing through the given points:

Given points: $(x_1, y_1) = \left(\frac{1}{2}, 5\right)$ and $(x_2, y_2) = \left(5\frac{1}{2}, 10\right)$ .
Use the formula for the slope: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 5}{5.5 - 0.5} = \frac{5}{5} = 1.$
The slope $m = 1$ is positive, indicating the line is increasing from left to right.

Hence, the function is a bottom-up function, indicating it increases as the x-values increase.

Therefore, the correct answer is: Bottom-up function.

Final Answer

Bottom-up function

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use m = (y₂ - y₁) / (x₂ - x₁) for two points
Mixed Numbers: Convert 5½ to 5.5 for easier calculation
Function Type: Positive slope means increasing (bottom-up) function ✓

Common Mistakes

Avoid these frequent errors

Confusing x and y coordinates when calculating slope
Don't subtract x-coordinates in numerator and y-coordinates in denominator = completely wrong slope! This mixes up the rise and run, giving meaningless results. Always use (y₂ - y₁) in numerator and (x₂ - x₁) in denominator.

Practice Quiz

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

FAQ

Everything you need to know about this question

How do I handle mixed numbers like 5½ in coordinates?

Convert mixed numbers to decimals for easier calculation! $5\frac{1}{2} = 5.5$ and $\frac{1}{2} = 0.5$ . This makes the slope formula much simpler to work with.

What's the difference between bottom-up and decreasing functions?

A bottom-up function means increasing (positive slope) - as x increases, y increases too. A decreasing function has negative slope - as x increases, y decreases. Don't confuse the terms!

Why does positive slope = 1 make this a bottom-up function?

When slope is positive, the line goes up from left to right. Since m = 1 > 0, this is an increasing or "bottom-up" function. The line rises as you move right.

Do I always need to simplify the slope fraction?

Yes! Always simplify your slope to lowest terms. In this problem, $\frac{5}{5} = 1$ , which clearly shows the slope is exactly 1.

Can a linear function be constant if it passes through different points?

No! A constant function has slope = 0, meaning all points have the same y-value. Since our points have different y-values (5 and 10), this cannot be constant.

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