Linear Function Analysis: Finding Positive and Negative Regions at x=7

Linear Function Signs with X-intercept Analysis

Given the function of the graph.

What are the areas of positivity and negativity of the function?

xy7

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 What are the positive and negative domains of the function?
00:03 and negative when the function is below the X-axis
00:08 The function is positive when it's above the X-axis
00:12 Let's identify when the function intersects the X-axis
00:15 We'll identify when the function is positive and when it's negative
00:28 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

Given the function of the graph.

What are the areas of positivity and negativity of the function?

xy7

2

Step-by-step solution

When we are asked what the domains of positivity of the function are, we are actually being asked at what values of X the function is positive: it is above the X-axis.

At what values of X does the function obtain positive Y values?

 

In the given graph, we observe that the function is above the X-axis before the point X=7, and below the line after this point. That is, the function is positive when X>7 and negative when X<7,

And this is the solution!

 

3

Final Answer

Positive 7>x 7 > x

Negative 7<x 7 < x

Key Points to Remember

Essential concepts to master this topic
  • Positivity Rule: Function is positive when graph is above x-axis
  • Reading Technique: At x=7 line crosses axis, left side positive, right negative
  • Verification: Check specific points: when x=5, y is positive; when x=9, y is negative ✓

Common Mistakes

Avoid these frequent errors
  • Confusing x-values with y-values when describing function signs
    Don't say 'positive when y>7' = wrong variable focus! This confuses the input (x) with the output (y). Always describe positivity using x-values: 'positive when x<7' means the function output is positive for those x-inputs.

Practice Quiz

Test your knowledge with interactive questions

Look at the linear function represented in the diagram.

When is the function positive?

–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333000

FAQ

Everything you need to know about this question

What does 'areas of positivity' actually mean?

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Areas of positivity are the x-values where the function gives positive y-values. Look at where the graph sits above the x-axis - those are your positive regions!

How do I read the graph to find where it's positive or negative?

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Follow the line from left to right! When the line is above the x-axis, the function is positive. When it's below the x-axis, it's negative. The crossing point at x=7 is where it changes sign.

Why is the function positive when x<7 instead of x>7?

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Look at the graph carefully! The line starts high on the left (positive y-values) and slopes downward, crossing the x-axis at x=7. So for x-values less than 7, the function is above the axis (positive).

What happens exactly at x=7?

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At x=7, the function equals zero because that's where the line crosses the x-axis. This point is neither positive nor negative - it's the boundary between the positive and negative regions.

How can I double-check my answer?

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Pick test points! Choose an x-value less than 7 (like x=5) and see if the graph shows a positive y-value there. Then pick an x-value greater than 7 (like x=9) and check if it's negative. This confirms your regions! ✓

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