Linear Function Through Points (0,5) and (6,5): Analyzing Horizontal Lines

Q: What's the difference between zero slope and undefined slope?

Zero slope means the line is horizontal (flat) - like $ y = 5 $. Undefined slope means the line is vertical (straight up and down) - like $ x = 3 $.

Q: How do I know if a line is a constant function?

Check if all the y-values are the same! If points like (0,5) and (6,5) have identical y-coordinates, then y always equals that same number - making it constant.

Q: Why is the slope zero when both y-values are 5?

Because slope measures vertical change over horizontal change. When y-values don't change (5 to 5), the vertical change is 0, so $ \frac{0}{6} = 0 $!

Q: What does a constant function look like on a graph?

A constant function appears as a perfectly horizontal line. No matter how far left or right you go, the height (y-value) stays exactly the same.

Q: Can I have a constant function with any y-value?

Yes! Constant functions can be $ y = 5 $, $ y = -2 $, $ y = 0 $, or any number. The key is that y never changes regardless of the x-value.

Horizontal Lines with Zero Slope

The graph of the linear function passes through the points $B(0,5),A(6,5)$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The graph of the linear function passes through the points $B(0,5),A(6,5)$

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Identify the points given: $B(0,5)$ and $A(6,5)$ .
Step 2: Use the slope formula to calculate the slope $m$ between these points:

The slope $m$ is calculated as:

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 5}{6 - 0} = \frac{0}{6} = 0

Since the slope $m = 0$ , the line is horizontal.

Step 3: Classify the line as a "constant function" because the slope is zero, indicating no increase or decrease.

The line is therefore a constant function where the $y$ -value remains at 5 for all $x$ -values.

Therefore, the answer is constant function.

Final Answer

Constant function

Key Points to Remember

Essential concepts to master this topic

Slope Formula: When y-values are equal, slope equals zero
Technique: Calculate $m = \frac{5-5}{6-0} = \frac{0}{6} = 0$
Check: Horizontal line means constant y-value for all x-values ✓

Common Mistakes

Avoid these frequent errors

Confusing horizontal and vertical lines
Don't think horizontal lines have undefined slope = vertical line properties! Horizontal lines have zero slope (flat), while vertical lines have undefined slope. Always remember: horizontal = zero slope, vertical = undefined slope.

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

What's the difference between zero slope and undefined slope?

Zero slope means the line is horizontal (flat) - like $y = 5$ . Undefined slope means the line is vertical (straight up and down) - like $x = 3$ .

How do I know if a line is a constant function?

Check if all the y-values are the same! If points like (0,5) and (6,5) have identical y-coordinates, then y always equals that same number - making it constant.

Why is the slope zero when both y-values are 5?

Because slope measures vertical change over horizontal change. When y-values don't change (5 to 5), the vertical change is 0, so $\frac{0}{6} = 0$ !

What does a constant function look like on a graph?

A constant function appears as a perfectly horizontal line. No matter how far left or right you go, the height (y-value) stays exactly the same.

Can I have a constant function with any y-value?

Yes! Constant functions can be $y = 5$ , $y = -2$ , $y = 0$ , or any number. The key is that y never changes regardless of the x-value.

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