Problem 63 - Calculus

Question

Give an example of: A linear function with a positive slope and a negative \(x\) intercept.

Step-by-Step Solution

Answer

The linear function \( y = 2x + 4 \) has a positive slope and a negative x-intercept.

Step 1: Understand the Requirements

We need a linear function, which is typically in the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. The function should have a positive slope and a negative x-intercept.

Step 2: Identify the Components of the Function

Since the slope \( m \) must be positive, we choose a positive number for \( m \), such as 2. For the linear function to have a negative x-intercept, the y-intercept \( c \) must be negative enough to allow the line to intersect the x-axis at a negative x.

Step 3: Construct the Function

With \( m = 2 \) and \( c = -3 \), a candidate function is \( y = 2x - 3 \). We will verify if this satisfies the conditions.

Step 4: Find the X-Intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \): \( 0 = 2x - 3 \). Add 3 to both sides to get \( 2x = 3 \). Divide by 2 to solve for \( x \), which gives \( x = \frac{3}{2} \). This value does not meet the requirement; re-select \( c \) to meet conditions.

Step 5: Correct the Function

We need the x-intercept to be negative. Let's choose \( c = 4 \). Thus, the function is \( y = 2x + 4 \).

Step 6: Verify X-Intercept of New Function

Set \( y = 0 \) in the function \( y = 2x + 4 \) and solve for \( x \): \( 0 = 2x + 4 \), which simplifies to \( 2x = -4 \). Divide by 2 to get \( x = -2 \). This satisfies the condition of a negative x-intercept.

Step 7: Confirm Solution

The function \( y = 2x + 4 \) has a positive slope \( m = 2 \) and a negative x-intercept at \( x = -2 \). Both conditions are satisfied.

Key Concepts

Slope

The slope of a linear function is a key mathematical concept that defines the rate at which one variable changes in relation to another. It shows how steep or flat the line is on a graph. In the context of a linear equation in the form \( y = mx + c \), the coefficient \( m \) represents the slope. A positive slope, like in our example function \( y = 2x + 4 \), means that as \( x \) increases, \( y \) also increases. This is often visualized as a line that rises as it moves from left to right.
A helpful way to understand slope is to think of it as "rise over run":

• "Rise" refers to the change in \( y \).
• "Run" refers to the change in \( x \).

If you were climbing a hill that has a slope of 2, you'd go up 2 meters for every 1 meter you walk horizontally. Remember, the greater the positive number, the steeper the line; for instance, a line with a slope of 10 is much steeper than one with a slope of 2.

X-intercept

The x-intercept of a linear function is the point where the line crosses the x-axis. This occurs when \( y = 0 \), which means the line has neither ascended nor descended from the x-axis.

To find the x-intercept of a function like \( y = 2x + 4 \), you set \( y \) to 0 and solve the equation:

\[ 0 = 2x + 4 \]

To solve for \( x \), subtract 4 from both sides to obtain \( 2x = -4 \), then divide by 2:

\[ x = -2 \]

This calculation determines that the x-intercept is at \( x = -2 \). In simpler terms, it tells us that the line meets the x-axis at the point (-2, 0). The value of the x-intercept can be crucial in understanding the graph's behavior, such as identifying where on the axis the line will cut across horizontally.

Y-intercept

The y-intercept is the point where the line of a linear function crosses the y-axis. It occurs when \( x = 0 \), meaning the line will intersect the y-axis at this precise location. The y-intercept can be straightforwardly identified from the linear equation \( y = mx + c \), where \( c \) represents the y-intercept.
For the function \( y = 2x + 4 \), the y-intercept is \( 4 \). This simply means that when \( x \) equals 0, \( y \) equals 4, so the graph crosses the y-axis at the point (0, 4).

Understanding the y-intercept helps in sketching the overall position and orientation of the line. Knowing both the y-intercept and the slope gives a complete picture of the line on a graph, demonstrating where it begins its ascent or descent relative to the y-axis.