Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

find the slope of the secant line passing through points \(P\) and \(Q\), where \(P\) and \(Q\) are points of the graph of \(f(x)\) with the indicated \(x\) -coordinates. \(f(x)=a x^{2}+b x+c ;\) the \(x\) -coordinates of \(P\) and \(Q\) are \(x=k\) and \(x=k+h(h \neq 0)\), respectively.

Step-by-Step Solution

Answer

The slope of the secant line passing through the points \(P\) and \(Q\) on the graph of the function \(f(x)=ax^{2}+bx+c\) where \(P\) and \(Q\) are the points at \(x=k\) and \(x=k+h\), respectively, is \(m=2ak+b+ah\).

Step 1: Substitution

First, let's evaluate the function \(f\) at points \(x=k\) and \(x=k+h\). Now, we have \(f(k)=ak^{2}+bk+c\) and \(f(k+h)=a(k+h)^{2}+b(k+h)+c\).

Step 2: Simplify \(f(k+h)\)

We simplify \(f(k+h)\) which would allow us to subtract \(f(k)\) more easily. Thus, we have \(f(k+h)=a(k^{2}+2hk+h^{2})+b(k+h)+c = ak^{2}+2ahk+ah^{2}+bk+bh+c\).

Step 3: Find the Difference

Next, find \(f(k+h)-f(k)\). This gives us the difference of the y-coordinates of points \(Q\) and \(P\), which is part of the slope formula. Thus, we have \(f(k+h)-f(k)= (ak^{2}+2ahk+ah^{2}+bk+bh+c) - (ak^{2}+bk+c) = 2ahk+ah^{2}+bh\).

Step 4: Find the Slope of the Secant Line

Finally, we find the slope \(m\) of the secant line using the formula \(m = \frac{f(k+h)-f(k)}{h}\). Substituting the results from the previous step, we find \(m=2ak+b+a h\).

Key Concepts

Secant Line

A secant line is an important concept in calculus. It's a straight line that intersects two points on a curve. This concept is very useful when you're trying to understand the rate of change between those two points.
In the context of a graph of a function, the secant line gives us an average rate of change between two distinct points.
When working with a quadratic function, like in the original problem, our goal is to find the slope of this secant line. This entails calculating the change in the y-values over the change in the x-values between the two points.

• Think of it like measuring the steepness of a hill between two points along a path.
• The slope of a secant line provides a 'snapshot' of how fast or slow the curve is changing between the two selected points.

Understanding this concept is crucial because it lays the groundwork for more advanced ideas, such as tangents and derivatives.

Slope Formula

The slope formula is a fundamental tool for finding how steep a line is. When dealing with a secant line, we use this formula to determine the line's slope.
The formula for the slope of a line is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]For a secant line passing through points \((x_1, y_1)\) and \((x_2, y_2)\), this formula calculates the change in the y-values and divides it by the change in the x-values.
In our exercise, the points \(P\) and \(Q\) are on the graph of a function, so the slope of the secant line becomes:
\[ m = \frac{f(k+h)-f(k)}{h}\]With quadratic functions, this means substituting the expressions for \(f(k)\) and \(f(k+h)\) into the formula to find the specific rate of change between those points. Here, you perform algebraic manipulations to get a clearer understanding of how the output (or y-value) changes with respect to \(x\).

Quadratic Functions

Quadratic functions are a specific type of polynomial function characterized by the highest power of the variable being 2. This means that any quadratic function can be expressed in the standard form:
\[ f(x) = ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
The graph of a quadratic function is a curve known as a parabola, which can either open upwards or downwards. Understanding the structure of quadratic functions helps you visualize how any changes to the x-values (like moving from \(k\) to \(k+h\)) affect the y-values.

• These functions are highly symmetrical and exhibit predictable patterns in their rate of change.
• Their symmetry can aid in simplifying calculations related to average rates of change or slopes of secant lines.

In the exercise, simplifying and understanding the form of \(f(k+h)\) is necessary for calculating the slope. Breaking it down shows how each term contributes to the overall structure of the parabola.