Problem 37 - Algebra 1

Question

Graph the function. $$ f(x)=-\frac{1}{2} x+1 $$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=-\frac{1}{2} x+1\) is a straight line with a negative slope that passes through points (0,1) and (2,0).

Step 1: Identify the Slope and Y-Intercept

The equation is given as \(f(x)=-\frac{1}{2} x+1\). Comparing this with the standard form of a linear equation \(y = mx + c\), we find that the slope 'm' is \( -\frac{1}{2} \) and the y-intercept 'c' is 1.

Step 2: Plot the Y-intercept

On the graph, mark a point at the y-intercept which is 1. This point is on the y-axis. This gives us the starting point for the line.

Step 3: Use the Slope to Plot the next point

The slope of the line is \( - \frac{1}{2} \), so we will go down 1 unit and over to the right 2 units from the y-intercept to plot the next point because the slope is negative. The negative sign of the slope indicates a downward trend.

Step 4: Draw the Line

Now, draw a straight line that passes through the points we have on the plot. This line represents the function \(f(x) = - \frac{1}{2}x + 1\).

Key Concepts

Slope

When we talk about the slope in the context of graphing linear functions, imagine it as a "steepness" factor of the line. The slope tells us how much the function increases or decreases as we move along the x-axis. In the given function, \(f(x) = -\frac{1}{2}x + 1\), the slope is \(-\frac{1}{2}\).

• The negative sign indicates that the line is sloping downwards as it moves from left to right.
• The fraction \(\frac{1}{2}\) suggests that for every 2 units we move to the right, the line goes down 1 unit.

This concept of slope is vital because it provides a visual representation of the rate of change. In simple terms, a line with a steep slope falls or rises quickly, while a flat slope represents slow change. Understanding slope helps you predict how fast or slow an outcome might happen based on changes in input.

Y-Intercept

The y-intercept is a critical point on a graph where the line crosses the y-axis. It's where the value of \(x\) is 0, making it a perfect point to use when starting to draw a graph. In the linear function \(f(x) = -\frac{1}{2}x + 1\), the y-intercept is 1.

• This point is written as \((0, 1)\) in coordinate form.
• The y-intercept provides a starting point for graphing the function, allowing us to build the rest of the line from there using the slope.

The y-intercept is an anchor point for the line on the graph. With the y-intercept known, you can begin to apply the slope to find more points and complete the linear representation of the function. This is the first point you plot when graphing a linear equation.

Linear Equations

Linear equations are fundamental to understanding how different variables relate to each other in a straight-line manner. These equations are usually written in the format \(y = mx + c\), where:

• \(m\) represents the slope.
• \(c\) is the y-intercept.

A linear equation graphs as a straight line due to its constant rate of change, signified by the slope. For example, in \(f(x) = -\frac{1}{2}x + 1\), each value of \(x\) yields a corresponding value of \(f(x)\), forming a line when connected.Linear equations are powerful tools because they can model and predict behavior over time or under varying conditions. They're widely used in everything from simple predictions to complex calculations, making understanding their graphing crucial in mathematics.