https://file.aiquickdraw.com/mathbot/file/1726794938683ottn4kxc.jpg

The problem states:

We are given a point \( P(2, 4) \) on the curve \( y = x^2 \), and another point \( Q \) that also lies on this curve, with \( Q \neq P \). The task is to express the slope of the line \( PQ \) as a function of the \( x \)-coordinate of point \( Q \), and sketch the graph of this slope function.

### Step 1: Coordinates of Points
- Point \( P \) has coordinates \( (2, 4) \), as given.
- For point \( Q \), its coordinates can be written as \( (x_Q, y_Q) \), where \( y_Q = x_Q^2 \), since \( Q \) also lies on the curve \( y = x^2 \).

### Step 2: Formula for the Slope of Line \( PQ \)
The slope \( m \) of the line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

In this case:
- \( (x_1, y_1) = (2, 4) \) (point \( P \))
- \( (x_2, y_2) = (x_Q, x_Q^2) \) (point \( Q \))

Thus, the slope of line \( PQ \) is:

\[
m(x_Q) = \frac{x_Q^2 - 4}{x_Q - 2}
\]

### Step 3: Simplifying the Expression
To simplify the slope function, we can factor the numerator:

\[
m(x_Q) = \frac{(x_Q - 2)(x_Q + 2)}{x_Q - 2}
\]

For \( x_Q \neq 2 \), the terms \( (x_Q - 2) \) cancel out, leaving:

\[
m(x_Q) = x_Q + 2 \quad \text{for} \quad x_Q \neq 2
\]

Thus, the slope of the line \( PQ \) is \( m(x_Q) = x_Q + 2 \).

### Step 4: Conclusion and Graph
- The function for the slope of the line \( PQ \) is \( m(x_Q) = x_Q + 2 \), which is a linear function.
- At \( x_Q = 2 \), the slope is undefined, as the two points coincide.

To sketch the graph of this function:
- The function is a straight line with a slope of 1 and a y-intercept at \( y = 2 \), excluding the point \( x_Q = 2 \) where the function is undefined.

Would you like more detailed steps or any additional clarifications?

---

### Follow-up Questions:
1. How do we handle the slope at \( x_Q = 2 \) mathematically?
2. Can you explain the process of factoring the numerator more clearly?
3. What is the geometric interpretation of the line \( PQ \) as \( x_Q \) approaches 2?
4. How would this problem change if the curve equation was different?
5. What are the potential applications of finding the slope of secant lines?

### Tip:
For problems involving secant lines, it helps to recall that as the two points \( P \) and \( Q \) get closer together, the slope of the secant line approaches the derivative of the function at that point.

Diberikan titik P(2, 4) pada kurva y = x^2. Titik Q juga terletak pada kurva tersebut dengan Q tidak sama dengan P. Nyatakan kemiringan garis PQ sebagai fungsi dalam absis titik Q dan sketsakan grafik dari fungsi kemiringan garis tersebut.

Learn how to find the slope of a secant line between points on the quadratic curve y = x^2. This problem explores key algebraic concepts such as the slope formula and quadratic functions. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation