Exploring Constant Function Creation with Two Points: Geometric Analysis

Q: What exactly makes a function constant?

A constant function always gives the same output (y-value) no matter what input (x-value) you use. It's like a horizontal line that never goes up or down!

Q: How can I tell if two points have the same y-coordinate from a graph?

Look at where each point aligns with the y-axis. If both points are at the same height above or below the x-axis, they have identical y-coordinates.

Q: Can a constant function pass through points with different x-values?

Absolutely! A constant function can connect any points that have the same y-coordinate, regardless of how far apart their x-values are.

Q: What would the equation look like if these points form a constant function?

The equation would be $ f(x) = c $, where c is the common y-coordinate of both points. For example, if both points have y = 3, then $ f(x) = 3 $.

Q: Is it possible for points to NOT form a constant function?

Yes! If the two points have different y-coordinates, then no constant function can pass through both of them. You'd need a different type of function instead.

Constant Functions with Identical Y-values

Is it possible to create a constant function with the two given points?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Can we create a constant function from 2 points?

00:04 Let's complete the points on the graph

00:07 It appears that the function is constant

00:10 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

Is it possible to create a constant function with the two given points?

Step-by-step solution

To solve this problem, we'll examine the two given points to see if a constant function can be defined through them.

A constant function is represented by $f(x) = c$ , where $c$ is a constant, meaning all $y$ -coordinates for the function must be the same regardless of $x$ .

Step 1: Identify the $y$ -coordinates of the given points from the plot.
Step 2: Check if these $y$ -coordinates are identical, indicating the output of the function does not change.
Step 3: If both $y$ -coordinates are the same, a constant function can indeed pass through both points.

Upon inspection, the $y$ -coordinates of the two points are the same, which satisfies the requirement for a constant function.

Therefore, it is possible to create a constant function using the two given points.

The correct conclusion is: Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Definition: Constant functions have form $f(x) = c$ with same y-value everywhere
Technique: Check if both points have identical y-coordinates on the graph
Check: Both points lie on same horizontal line through y-axis ✓

Common Mistakes

Avoid these frequent errors

Focusing only on x-coordinates instead of y-coordinates
Don't check if x-values are the same = wrong analysis! Constant functions can have different x-values but must have identical y-values. Always examine the y-coordinates of both points to determine if they're equal.

Practice Quiz

Test your knowledge with interactive questions

Does the function in the graph decrease throughout?

FAQ

Everything you need to know about this question

What exactly makes a function constant?

A constant function always gives the same output (y-value) no matter what input (x-value) you use. It's like a horizontal line that never goes up or down!

How can I tell if two points have the same y-coordinate from a graph?

Look at where each point aligns with the y-axis. If both points are at the same height above or below the x-axis, they have identical y-coordinates.

Can a constant function pass through points with different x-values?

Absolutely! A constant function can connect any points that have the same y-coordinate, regardless of how far apart their x-values are.

What would the equation look like if these points form a constant function?

The equation would be $f(x) = c$ , where c is the common y-coordinate of both points. For example, if both points have y = 3, then $f(x) = 3$ .

Is it possible for points to NOT form a constant function?

Yes! If the two points have different y-coordinates, then no constant function can pass through both of them. You'd need a different type of function instead.

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