https://file.aiquickdraw.com/mathbot/file/1726000464763aicq961b.jpg

To determine which statement is **true** about the linear functions A and B, let's analyze the information given in the table and graph.

### **Function A (from the table):**
The table provides pairs of \((x, y)\) values for function A.

| \(x\) | \(-6\) | \(-2\) | 2  | 4  |
|------|--------|--------|----|----|
| \(y\) | \(-5\) | \(-3\) | \(-1\) | 0  |

#### Step 1: Find the slope of Function A
To find the slope \(m\), we use the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-6, -5)\) and \((-2, -3)\), we calculate:
\[
m = \frac{-3 - (-5)}{-2 - (-6)} = \frac{-3 + 5}{-2 + 6} = \frac{2}{4} = \frac{1}{2}
\]
So, the slope of Function A is \( \frac{1}{2} \), which is positive.

#### Step 2: Find the y-intercept of Function A
The y-intercept is the value of \(y\) when \(x = 0\). From the table, when \(x = 0\), \(y = 0\). Thus, the y-intercept of Function A is \(0\).

### **Function B (from the graph):**
From the graph of Function B, we can visually identify that it has a **negative slope** because it slopes downward from left to right. We can also see that the graph crosses the y-axis at \((0, 4)\), meaning the y-intercept of Function B is \(4\).

#### Step 3: Find the slope of Function B
Using two points on the graph, say \((0, 4)\) and \((4, -4)\), we calculate the slope \(m\):
\[
m = \frac{-4 - 4}{4 - 0} = \frac{-8}{4} = -2
\]
Thus, the slope of Function B is \(-2\), which is negative.

### **Now, let's review the options:**
1. **The y-intercepts of the functions have opposite signs** – False. The y-intercept of Function A is 0, and for Function B, it's 4. These are not opposite in sign.
2. **The slope of each function is negative** – False. The slope of Function A is positive \(\frac{1}{2}\), while the slope of Function B is negative \(-2\).
3. **The y-intercept of Function A is farther from the origin than the y-intercept of Function B** – False. The y-intercept of Function A is \(0\), which is closer to the origin than the y-intercept of Function B (\(4\)).
4. **The slope of Function A is less steep than the slope of Function B** – True. The slope of Function A is \(\frac{1}{2}\), which is less steep than the slope of Function B, which is \(-2\).

### **Conclusion:**
The correct statement is: **The slope of Function A is less steep than the slope of Function B**.

---

Would you like further details on any of these steps? Here are some related questions:

1. How can we further analyze the relationship between slopes and steepness of functions?
2. Why does the sign of the slope affect the direction of the line on the graph?
3. What happens when the y-intercept is zero in a function?
4. How do changes in the x-values affect the calculation of the slope in a linear function?
5. Can we compare the rate of change between two linear functions based solely on their slopes?

**Tip:** Always remember that a more negative slope results in a steeper downward slope, while a more positive slope results in a steeper upward slope.

Linear functions A and B are represented by the table and graph shown. Which statement is true about the functions?

Explore the comparison between linear functions A and B by analyzing their slopes and y-intercepts. Learn how to calculate the slope and determine the steepness of each function using the slope formula. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation