https://file.aiquickdraw.com/mathbot/file/1727829732700fq99fenr.jpg

can you explain how this person solved this step by step

Certainly! I'll walk you through each step of the solution in the image you provided. This is a detailed process for finding the slope (denoted by \( m \)) of the secant line between two points on a curve.

### **Step-by-step Explanation:**

1. **Given Points:**
   The points provided are \( (-4, 56) \) and \( (-4 + h, k(-4 + h)) \). This indicates two points on the graph, with one point being fixed at \( (-4, 56) \) and the other point shifting based on the variable \( h \).

2. **Slope Formula:**
   The slope \( m \) is calculated as:
   \[
   m = \frac{k(-4+h) - 56}{(-4+h) - (-4)}
   \]
   This is the standard slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where the second point is \( (-4+h, k(-4+h)) \) and the first point is \( (-4, 56) \).

3. **Simplifying the Denominator:**
   The denominator simplifies as:
   \[
   (-4+h) - (-4) = h
   \]
   So the slope equation becomes:
   \[
   m = \frac{k(-4+h) - 56}{h}
   \]

4. **Substituting the Function \( k(x) \):**
   Based on the problem setup, \( k(x) = 4x^2 + 2x \). So, for \( k(-4+h) \), we substitute into this function:
   \[
   k(-4+h) = 4(-4+h)^2 + 2(-4+h)
   \]

5. **Expanding the Expression \( k(-4+h) \):**
   First, expand \( (-4+h)^2 \):
   \[
   (-4+h)^2 = 16 - 8h + h^2
   \]
   Now, substitute this into the function \( k(-4+h) \):
   \[
   k(-4+h) = 4(16 - 8h + h^2) + 2(-4+h)
   \]
   Simplify the terms:
   \[
   k(-4+h) = 64 - 32h + 4h^2 - 8 + 2h = 56 - 30h + 4h^2
   \]

6. **Substituting Back into the Slope Formula:**
   Now, substitute \( k(-4+h) = 56 - 30h + 4h^2 \) into the slope formula:
   \[
   m = \frac{(56 - 30h + 4h^2) - 56}{h}
   \]
   This simplifies to:
   \[
   m = \frac{-30h + 4h^2}{h}
   \]

7. **Simplifying the Slope Expression:**
   Factor out \( h \) from the numerator:
   \[
   m = \frac{h(-30 + 4h)}{h}
   \]
   The \( h \)'s cancel out:
   \[
   m = -30 + 4h
   \]

Thus, the final expression for the slope \( m \) is:
\[
m = -30 + 4h
\]

### **Summary of Steps:**
1. Start with two points \( (-4, 56) \) and \( (-4+h, k(-4+h)) \).
2. Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
3. Simplify the denominator to \( h \).
4. Substitute the function \( k(x) = 4x^2 + 2x \) and expand for \( k(-4+h) \).
5. Simplify and factor the expression.
6. Cancel out \( h \) to get the final slope.

---

Do you want more details on any of these steps, or do you have any questions?

Here are five related questions:
1. What is the geometric meaning of the slope of a secant line?
2. How would this process change if the function \( k(x) \) were different?
3. What is the limit of the slope \( m \) as \( h \to 0 \)?
4. How does this process relate to the concept of the derivative?
5. Can this method be applied to find the slope of a tangent line?

**Tip:** When simplifying algebraic expressions, be careful to expand and combine like terms correctly before factoring.

Can you explain how this person solved this step by step?

Step-by-step explanation of how to find the slope of a secant line between two points on a quadratic curve, using algebraic simplification techniques. Suitable for high school students studying advanced algebra and precalculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation