Problem 12 - Precalculus with Limits

Question

In Exercises 11-18, (a) write the linear function \(f\) such that it has the indicated function values and (b) sketch the graph of the function. \(f(-3) = -8\), \(f(1) = 2\)

Step-by-Step Solution

Answer

The linear function that fits the given conditions is \(f(x) = 2x - 2\). The graph of the function will be a straight line crossing the y-axis at -2 and having a slope of 2.

Step 1: Calculation of Slope

Given two coordinates \((-3, -8)\) and \( (1, 2)\), we'll use the slope formula \(m = \frac{y2 - y1}{x2 - x1}\) where \(x1 = -3\), \(y1 = -8\), \(x2 = 1\) and \(y2 = 2\). This gives us \(m = \frac{2 - (-8)}{1 - (-3)} = 2\).

Step 2: Calculation of y-Intercept

We'll use the formula \(y = mx + c\), substituting the values for m, x and y to find c (y-intercept). We can take \((-3, -8)\) to find the y-intercept. \(so -8 = 2(-3) + c, \implies c = -8 + 6 = -2\). So the y-intercept is -2.

Step 3: Formulate Linear Function

With the slope \(m = 2\) and y-intercept \(c = -2\), we can now write down the linear function as \(f(x) = 2x - 2\).

Step 4: Sketch the graph

To make a graph, first make a point at y-intercept (-2 on y axis) then, as slope is 2, for every 1 point move in positive x direction, we move 2 points up on y axis. connect the points to create a continuous straight line representing the linear function.

Key Concepts

Slope Calculation

To find the slope of a linear function, we use the formula for calculating the slope, which is given by:

• The slope \(m = \frac{y2 - y1}{x2 - x1}\)
• This means you subtract the first y-value from the second y-value, and the first x-value from the second x-value, then divide both results.

Having a slope means you understand how much the line rises vertically compared to how much it runs horizontally.
Each linear function is characterized by a consistent slope, allowing us to predict behavior across any two points.

Y-Intercept

The y-intercept is a critical component of any linear function. It tells us where the line crosses the y-axis. To calculate it, use the equation of a line, which is:

• The standard form is \(y = mx + b\).
• To solve for \(b\), choose any point on the line (we used \((-3, -8)\) in the exercise).
• Substitute the values into the equation \(-8 = 2(-3) + b\) and solve for \(b\).

Knowing the y-intercept helps in sketching and defining the linear function.

Graph Sketching

Sketching the graph of a linear function begins with plotting the y-intercept on the graph. This is the point where the line crosses the y-axis, in this case at \(y = -2\).
After marking the intercept:

• Use the slope, which we calculated earlier as 2, to find another point. This means that for every 1 unit you move to the right along the x-axis, you move 2 units up along the y-axis.
• Connect these points to form a straight line, extending it across the graph.

Remember, in linear functions, this relationship remains constant, ensuring the line is straight throughout.