Linear Function Through Points (0,0) and (4,6): Graph Analysis

Q: What does 'bottom-up function' actually mean?

A bottom-up function is another way to say increasing function. As you move from left to right on the graph, the line goes upward, like climbing from bottom to top!

Q: Why isn't this a constant function?

A constant function has a slope of zero and forms a horizontal line. Since our slope is $ \frac{3}{2} $, the line is slanted upward, not flat.

Q: How do I remember which coordinate goes where in the slope formula?

Think 'rise over run': the y-values (vertical change) go on top, x-values (horizontal change) go on bottom. Always subtract the same point's coordinates: $ \frac{y_2-y_1}{x_2-x_1} $.

Q: What if I get a negative slope instead?

A negative slope means the function is decreasing - the line goes down from left to right. The steeper the negative slope, the faster it decreases.

Q: Does it matter which point I call (x₁,y₁) and which is (x₂,y₂)?

No, it doesn't matter! As long as you're consistent and subtract the same point's coordinates, you'll get the same slope. Try it both ways with our points - you'll get $ \frac{3}{2} $ either way.

Slope Analysis with Point Coordinates

The graph of the linear function passes through the points $B(0,0),A(4,6)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's first determine the type of slope we're dealing with.

00:11 Next, we'll find the slope using two points. It's simple! Let's pick those points.

00:23 Now, we'll use the slope formula. Remember, the formula is: Y two minus Y one, over X two minus X one.

00:32 Great! Let's substitute the right values into the formula and solve to find the slope.

00:51 Awesome! Since the slope is positive, it tells us the function is increasing.

00:58 And that's how we solve the problem! Well done!

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 $B(0,0),A(4,6)$

Step-by-step solution

To determine the type of linear function represented by a line passing through the points $B(0,0)$ and $A(4,6)$ , we follow these steps:

Step 1: Identify the points. Here, we have $B(0,0)$ and $A(4,6)$ .
Step 2: Use the slope formula to find the slope of the line. The formula to find the slope $m$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ .

Plug in the coordinates of the points:

$m = \frac{6 - 0}{4 - 0} = \frac{6}{4} = \frac{3}{2}$ .

Step 3: Analyze the slope: Since the slope $m = \frac{3}{2}$ is positive, the function is an increasing function.
Step 4: Determine the type of function: A positive slope indicates that the function is increasing as we move from left to right on the graph.

Therefore, the correct description of this linear function, based on the given options, is a Bottom-up function.

Final Answer

Bottom-up function

Key Points to Remember

Essential concepts to master this topic

Slope Formula: Use $m = \frac{y_2 - y_1}{x_2 - x_1}$ to find rate of change
Calculation: $m = \frac{6-0}{4-0} = \frac{3}{2}$ gives positive slope
Check: Positive slope means y increases as x increases, confirming increasing function ✓

Common Mistakes

Avoid these frequent errors

Confusing slope sign with function behavior
Don't assume negative coordinates mean decreasing function = wrong classification! The slope sign determines behavior, not the coordinates themselves. Always calculate the slope first, then interpret: positive slope = increasing (bottom-up), negative slope = decreasing.

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

What does 'bottom-up function' actually mean?

A bottom-up function is another way to say increasing function. As you move from left to right on the graph, the line goes upward, like climbing from bottom to top!

Why isn't this a constant function?

A constant function has a slope of zero and forms a horizontal line. Since our slope is $\frac{3}{2}$ , the line is slanted upward, not flat.

How do I remember which coordinate goes where in the slope formula?

Think 'rise over run': the y-values (vertical change) go on top, x-values (horizontal change) go on bottom. Always subtract the same point's coordinates: $\frac{y_2-y_1}{x_2-x_1}$ .

What if I get a negative slope instead?

A negative slope means the function is decreasing - the line goes down from left to right. The steeper the negative slope, the faster it decreases.

Does it matter which point I call (x₁,y₁) and which is (x₂,y₂)?

No, it doesn't matter! As long as you're consistent and subtract the same point's coordinates, you'll get the same slope. Try it both ways with our points - you'll get $\frac{3}{2}$ either way.

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