Identify the Correct Function: Analyzing 4 Linear Graphs on Coordinate Plane

Q: How can I tell if a line has slope 1 just by looking?

Look for a 45-degree angle! If the line makes equal angles with both axes and goes up-right at a gentle slope, it's likely $ y = x $. The line should pass through points like (1,1), (2,2), etc.

Q: What's the difference between y = x and y = 6x visually?

$ y = 6x $ is much steeper! It rises 6 units up for every 1 unit right, while $ y = x $ rises just 1 unit up for 1 unit right. Line 4 is gentle, not steep.

Q: Why does the line pass through the origin?

When there's no y-intercept term (like +3 or -5), the line must pass through (0,0). Functions like $ y = mx $ always start at the origin.

Q: How do I know it's not y = -x?

Check the direction! $ y = -x $ slopes downward from left to right (negative slope), but line 4 slopes upward from left to right (positive slope).

Q: Can I use two points to find the slope?

Absolutely! Pick any two clear points on line 4 and use $ m = \frac{y_2 - y_1}{x_2 - x_1} $. Since it passes through (0,0) and appears to go through (1,1), the slope is $ \frac{1-0}{1-0} = 1 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Match the function to line 4

00:04 The function is increasing, therefore the slope is positive

00:09 We'll eliminate all functions with negative slope

00:33 We know that this function matches graph 3

00:36 Therefore we'll choose the only remaining matching function

00:39 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Choose the appropriate function for the number line 4.

2

Step-by-step solution

To solve this problem, we need to determine the function that corresponds to the line labeled "4" on the number line.

The line labeled "4" is depicted in orange and appears to pass through the origin, indicating that the y-intercept is 0. Therefore, the equation has the form $y = mx$ .

Upon examining this line, we observe that it is direct and passes through the origin at a $45^\circ$ angle to both axes, suggesting that the slope $m$ is 1. This is to say, for each unit increase in $x$ , $y$ also increases by the same magnitude.

The equation thus simplifies to $y = x$ .

We next compare our result to the given function choices:

Choice A: $y = -x$ (Negative slope)
Choice B: $y = x$ (Positive slope of 1)
Choice C: $y = 6x$ (Steeper positive slope)
Choice D: $y = -6x$ (Steeper negative slope)

The line labeled "4" directly matches Choice B, $y = x$ , as both the direction and slope align with our observations.

Therefore, the function corresponding to the number line labeled "4" is $y = x$ .

3

Final Answer

$y=x$

Key Points to Remember

Essential concepts to master this topic

Rule: Line through origin has form $y = mx$ where m is slope
Technique: 45° angle to axes means slope = 1, so $y = x$
Check: Orange line 4 passes through (1,1) and (2,2): confirms $y = x$ ✓

Common Mistakes

Avoid these frequent errors

Confusing line direction with slope value
Don't assume steeper lines always have larger slopes like 6x = steeper than x! Line 4 has a gentle 45° slope (m=1), not steep like 6x would be. Always check if the line rises 1 unit up for each 1 unit right.

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 can I tell if a line has slope 1 just by looking?

+

Look for a 45-degree angle! If the line makes equal angles with both axes and goes up-right at a gentle slope, it's likely $y = x$ . The line should pass through points like (1,1), (2,2), etc.

What's the difference between y = x and y = 6x visually?

+

$y = 6x$ is much steeper! It rises 6 units up for every 1 unit right, while $y = x$ rises just 1 unit up for 1 unit right. Line 4 is gentle, not steep.

Why does the line pass through the origin?

+

When there's no y-intercept term (like +3 or -5), the line must pass through (0,0). Functions like $y = mx$ always start at the origin.

How do I know it's not y = -x?

+

Check the direction! $y = -x$ slopes downward from left to right (negative slope), but line 4 slopes upward from left to right (positive slope).

Can I use two points to find the slope?

+

Absolutely! Pick any two clear points on line 4 and use $m = \frac{y_2 - y_1}{x_2 - x_1}$ . Since it passes through (0,0) and appears to go through (1,1), the slope is $\frac{1-0}{1-0} = 1$ .

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Identify the Correct Function: Analyzing 4 Linear Graphs on Coordinate Plane

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