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Chapter 9: Problem 35

GRAPHING FUNCTIONS Graph the function. $$f(x)=-x+1$$

Short Answer

Expert verified
The graph of the function \(f(x) = -x + 1\) is a straight line that intercepts the y-axis at the point (0,1) and has a negative slope of -1, so it decreases from left to right.

Step by step solution

01

Identify the Slope and Y-Intercept

The highest degree of the polynomial term for the function \(f(x)\) is 1, indicating that this is a linear function. A linear function is of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Comparing \(f(x)=-x+1\) with this form, one can see that the slope \(m\) is -1 and the y-intercept \(c\) is 1.
02

Draw the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. Here, the y-intercept is 1, so the first point will be at (0,1) on the y-axis. Make a dot at this point on the graph.
03

Apply the Slope

The slope, \(m=-1\), tells us that each step to the right on the x-axis will go one step down on the y-axis. This is due to the negative slope. Starting from the y-intercept, move one step to the right and one step down, this will be the second point at (1,0). Draw a line through these two points extending over the entire graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope
Understanding the concept of slope is crucial when working with linear functions. The slope (\( m\)) tells us how steep a line is and the direction it moves. It is calculated by the change in the y-value divided by the change in the x-value between two points on a line. In the context of the equation \( f(x) = -x + 1\):
  • Here, the slope (\( m\)) is \(-1\).
  • A negative slope means the line moves downwards as it goes from left to right across the graph.
  • This means each unit you move to the right on the x-axis, you move a unit down on the y-axis.
Slopes are a measurement of a line's rate of change and can tell you a lot about how one variable is related to another. It helps in understanding how quickly y increases or decreases as x increases.
Y-Intercept
The y-intercept is another vital concept to comprehend when working with linear equations. It refers to the point where the line intersects the y-axis. In the equation \( f(x) = -x + 1\):
  • The y-intercept (\( c\)) is \(1\).
  • This is the point \((0, 1)\), showing clearly where the line cuts through the y-axis.
Understanding the y-intercept makes it easier to start plotting your graph because it is a definite starting point. Once you identify this point, further plotting the line using the slope becomes easier.
Linear Equations
Linear equations are equations that form straight lines on a graph. These equations have no exponents higher than 1, meaning they graph as straight lines. They are generally in the format:\[y = mx + c\]where:
  • \( m\) is the slope.
  • \( c\) is the y-intercept.
For the function \( f(x) = -x + 1\):
  • It follows the standard linear equation form, where the slope is \(-1\) and the y-intercept is \(1\).
  • Any changes in these values will alter the line's steepness and position on the graph.
Linear equations are simple yet powerful in modeling relationships between two variables due to their constant rate of change.
Function Graphing
Graphing a linear function helps visualize the relationship between variables. When graphing, the following steps are useful:
  • Identify and plot the y-intercept first, as this shows your starting point on the graph.
  • Use the slope to determine the next point. This involves moving according to the slope's rise over run ratio.
  • Connect these points with a straight line extending in both directions.
In the case of \( f(x) = -x + 1\):
  • Start at the point \((0,1)\) on the y-axis.
  • Use the slope \(-1/1\) (down 1 unit, right 1 unit) to find the next point \((1,0)\).
  • Draw a straight line through these points, and the graph of your function is complete.
Function graphing is a versatile tool for visualizing and understanding linear relationships, making abstract algebraic equations tangible.

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