Problem 26 - Algebra Form and Function

Question

Identify the slope and \(y\) -intercept and graph the function. $$ f(x)=-2+0.5 x $$

Step-by-Step Solution

Answer

Answer: The slope of the function is \(0.5\), and the y-intercept is \(-2\). To graph the function, start by plotting the y-intercept at the point \((0, -2)\), then use the slope to find another point, such as \((1, -1.5)\), and connect the points to create the line representing the function.

Step 1: 1. Identify the slope and y-intercept

The given function is \(f(x) = -2 + 0.5x\). Comparing it to the slope-intercept form (\(y = mx + b\)), we can see that the slope (m) is \(0.5\) and the y-intercept (b) is \(-2\).

Step 2: 2. Graph the function

To graph the function, follow these steps: a. Plot the y-intercept: Begin by plotting the point where the line crosses the y-axis. In our case, it is at the point \((0, -2)\). b. Use the slope to find another point: The slope of \(0.5\) means that when \(x\) increases by \(1\), \(y\) increases by \(0.5\). So, starting from the point \((0, -2)\), move right along the x-axis by \(1\) unit and up along the y-axis by \(0.5\) units. This will give us the point \((1, -1.5)\). c. Draw the line: Connect the two points \((0, -2)\) and \((1, -1.5)\), and extend the line to the edges of your graph. The graph of the function \(f(x) = -2 + 0.5x\) should now be correctly displayed on the coordinate plane.

Key Concepts

Slope-Intercept Form

When graphing linear equations, one of the most straightforward ways to interpret them is through the slope-intercept form. This is represented as \( y = mx + b \). In this equation: the variable \( m \) represents the slope of the line, while \( b \) stands for the y-intercept.

The slope-intercept form provides a clear, direct way to visualize a line on a graph. It's easy to see where your line will cross the y-axis (at the y-intercept) and how steep the line will be (indicated by the slope).

For example, in the function \( f(x) = -2 + 0.5x \), we immediately identify that \( m = 0.5 \) and \( b = -2 \). This tells us that the line rises as it moves from left to right, and it crosses the y-axis at -2.

Slope

The slope is a fundamental concept when working with linear equations, as it indicates the steepness and direction of the line. Slope is calculated as the ratio of the change in the y-values to the change in the x-values, often noted as \( \frac{\Delta y}{\Delta x} \).

In the slope-intercept equation \( y = mx + b \), the \( m \) term represents the slope. A positive slope, such as \( 0.5 \), means that for every increase by 1 in x, y increases by 0.5. This results in an upward-sloping line across the graph. If the slope were negative, the line would decline as it moved from left to right.

Understanding slope is crucial for graphing, as it allows you to determine how to move from one point to another on the line. In our example, starting from the y-intercept \((0, -2)\), the slope dictates that moving 1 unit right in x results in moving 0.5 units up in y.

Y-Intercept

The y-intercept is the point where the graph of the equation crosses the y-axis. In the slope-intercept form \( y = mx + b \), the \( b \) coefficient specifies this intercept. It tells you the value of \( y \) when \( x \) is zero.

For our function, \( f(x) = -2 + 0.5x \), the y-intercept is -2. This means that the line touches the y-axis at the point \((0, -2)\). It's a handy starting point for graphing because it provides a definite point from which to plot the rest of your line.

Knowing the y-intercept allows you to quickly locate a starting point on your graph. From there, you employ the slope to find additional points, ensuring your line is accurately depicted.