Problem 21 - 5 Steps to A 5: AP Calculus AB 2021

Question

Given a linear function \(y=f(x)\), with \(f(2)=4\) and \(f(-4)=10,\) find \(f(x)\).

Step-by-Step Solution

Answer

Answer: The equation of the linear function is f(x) = -x + 6.

Step 1: Determine the slope

To find the slope, we will use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) with our two points \((x_1, y_1) = (2, 4)\) and \((x_2, y_2) = (-4, 10)\). \(m = \frac{10 - 4}{-4 - 2} = \frac{6}{-6} = -1\) The slope \(m = -1\).

Step 2: Find the equation

Now that we have the slope, we will use the slope-point equation \(y - y_1 = m(x - x_1)\), with the slope we found, and one of the given points (e.g., \((x_1, y_1) = (2, 4)\)): \(y - 4 = -1(x - 2) \Rightarrow y - 4 = -x + 2\) Now, we isolate the \(y\) term: \(y = -x + 2 + 4 \Rightarrow y = -x + 6\) Thus, the linear function is: \(f(x) = -x + 6\)

Key Concepts

Slope Calculation

The slope of a line is a fundamental component of linear functions. It tells us how steep or flat the line is and in which direction it goes. To calculate the slope, we use two points on the line, labeled

• o(\(x_1, y_1\)) and
• (\(x_2, y_2\)).To find the slope, denoted by \(m\), we use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula is sometimes called the "rise over run," where the "rise" is the change in \(y\)-values and the "run" is the change in \(x\)-values. For our problem, using the points (2, 4) and (-4, 10), the calculation would be \[m = \frac{10 - 4}{-4 - 2} = \frac{6}{-6} = -1\]So, the slope of the line is \(-1\). This means the line decreases at an angle going from left to right. A positive slope means a line angles upwards, while a negative slope, like ours, means it angles downwards.

Equation of a Line

The equation of a line in slope-intercept form is expressed as \(y = mx + b\). Here, \(m\) stands for the slope, and \(b\) is the y-intercept, which is where the line crosses the y-axis. To determine this equation when we know a point on the line and its slope, we use the point-slope form: \[y - y_1 = m(x - x_1)\]In our problem, we use one of the points, like (2, 4), and our previously calculated slope, \(-1\): \[y - 4 = -1(x - 2)\]Simplifying this equation step by step, we multiply the \(-1\) across the terms:\[y - 4 = -x + 2\]Next, we add 4 on both sides to solve for \(y\):\[y = -x + 2 + 4\]Which simplifies to:\[y = -x + 6\]Thus, the straight line equation of our function is \(f(x) = -x + 6\). This tells us at any point, just plug in the \(x\)-value to find the corresponding \(y\)-value.

Function Determination

Function determination involves finding the rule that defines the relationship between input \(x\) and output \(y\). For linear functions, this is often done using the relationship established in the equation \(y = mx + b\). After determining the slope and the equation of a line from given points, we identified our function as \(f(x) = -x + 6\). In this function:

• The slope \(-1\) tells us how \(y\) changes with \(x\).
• The y-intercept \(6\) provides the starting point when \(x = 0\).

This results in a linear depiction where for every unit increase in \(x\), \(y\) decreases by 1 unit, due to our slope of \(-1\). Knowing how to construct and understand this function allows us to easily predict any output values given input, sketch the line graph, and fundamentally understand the linear relationship in various real-world scenarios.