Problem 86 - Algebra A Combined Function

Question

Graph each linear function. \(f(x)=-x+1\)

Step-by-Step Solution

Answer

The graph of \(f(x) = -x + 1\) is a line with slope -1, crossing the y-axis at \((0, 1)\).

Step 1: Identify the slope and y-intercept

The given linear function is in the slope-intercept form, \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the function \(f(x) = -x + 1\), the slope \(m\) is \(-1\) and the y-intercept \(b\) is \(1\). This means the line crosses the y-axis at \((0, 1)\).

Step 2: Plot the y-intercept

On a graph, find and plot the y-intercept. The y-intercept is the point \((0, 1)\). Place a point on the graph where the line will cross the y-axis.

Step 3: Use the slope to find another point

The slope \(-1\) can be written as \(-1/1\). This means from the y-intercept \((0, 1)\), you move down 1 unit and right 1 unit to find the next point. This gives you the point \((1, 0)\) to plot on the graph.

Step 4: Draw the line

Using a ruler, draw a straight line through the points \((0, 1)\) and \((1, 0)\) extending it in both directions. This line represents the graph of the function \(f(x) = -x + 1\). Ensure the line continues beyond these points on both sides, as linear functions extend infinitely in both directions.

Key Concepts

Graphing Linear Equations

Graphing linear equations may sound complex at first, but it's all about finding a line on the graph that represents the equation. A linear equation forms a straight line, and each solution of the equation represents a point on this line. To graph a linear equation effectively, you need specific details to start such as the slope and the y-intercept. These are the tools that can help you place the initial points on the graph. Let's see how you can plot a linear function using these locations:

• First, identify important components of the equation: the slope and the y-intercept.
• Next, plot the y-intercept on the y-axis of your graph. This will be your starting point.
• Finally, use the slope to determine the direction and steepness of the line. Move from the y-intercept according to the slope, marking another point before you draw your line across the entire graph.

Understanding these parts will simplify your graphing journey, helping you accurately represent the given formula.

Slope-Intercept Form

The slope-intercept form is a standardized way to write linear equations and is incredibly useful for graphing. It's given as \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is helpful because it immediately tells you how to start plotting your points. With \( f(x) = mx + b \), you can:

• Easily spot what the slope and y-intercept are.
• Use these values to place your first point on the graph.
• Understand how the line behaves just by considering these two values: the slope ('m') determines the line's angle with the horizontal, and 'b' shows where it cuts the y-axis.

Let's recall, in our function \( f(x) = -x + 1 \), the slope \( m \) is \(-1\), and \( b \) is \(1\). This tells us the line will decline at a 45-degree angle and intersect the y-axis at \( (0, 1) \). Perfect for swiftly and accurately graphing!

Coordinate Plane

The coordinate plane is your playground for graphing linear equations. Think of it as a big grid made of an x-axis (horizontal line) and a y-axis (vertical line). All your plotting will happen here, and it helps in visualizing exactly what the equation looks like. When using the coordinate plane:

• Remember the x-axis runs left to right, and the y-axis runs bottom to top.
• Every point is in (x, y) format, where 'x' represents how far along the x-axis a point is, and 'y' represents how far up or down the y-axis it is.
• To graph a linear equation, you need to place points on this plane using the values from your slope and y-intercept.

In particular, the function we discussed \( f(x) = -x + 1 \) starts at the point \( (0, 1) \) on the y-axis, and then, using the slope \( -1 \) (down 1 unit and over 1 unit), another point \( (1, 0) \) is determined, allowing you to graph the line between these points.

Y-Intercept

The y-intercept is one of the most crucial aspects of a linear equation as it tells you the point where the line crosses the y-axis. In simple terms, it's the starting point for plotting your graph on the coordinate plane. This parameter, represented as 'b' in the slope-intercept form equation \( f(x) = mx + b \), gives you an immediate anchor point on the graph. To use the y-intercept effectively:

• Locate the point on the y-axis corresponding to the y-intercept value 'b'.
• Plot this coordinate starting from \( (0, b) \).
• This point does not move horizontally since it is exactly on the y-axis, at \( x = 0 \).

In our function \( f(x) = -x + 1 \), the y-intercept is \( 1 \). Hence, the graph begins at \( (0, 1) \). From here, you'll use the slope to find other points and extend the line in both directions to complete the graph of the equation.