Problem 103
Question
Graph the linear function fon a domain of \([-0.1,0.1]\) for the function whose slope is 75 and \(y\) -intercept is \(-22.5 .\) Label the points for the input values of \(-0.1\) and \(0.1 .\)
Step-by-Step Solution
Verified Answer
The points \((-0.1, -30)\) and \((0.1, -15)\) are labeled on the graph of the function.
1Step 1: Define the Linear Function
The general formula for a linear function is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Given, the slope \( m = 75 \) and the y-intercept \( b = -22.5 \), the linear function can be defined as \( y = 75x - 22.5 \).
2Step 2: Calculate the Output for x = -0.1
Substitute \( x = -0.1 \) into the linear function to find the corresponding \( y \)-value.\[y = 75(-0.1) - 22.5 = -7.5 - 22.5 = -30\]So, the point is \((-0.1, -30)\).
3Step 3: Calculate the Output for x = 0.1
Substitute \( x = 0.1 \) into the linear function:\[y = 75(0.1) - 22.5 = 7.5 - 22.5 = -15\]So, the point is \((0.1, -15)\).
4Step 4: Plot the Points on the Graph
Plot the points \((-0.1, -30)\) and \((0.1, -15)\) on the coordinate plane. Connect these points with a straight line, as this is a linear function, to visualize the line.
5Step 5: Verify and Label the Points
Ensure the points are accurately plotted, reflecting their calculated coordinates. Label the point at \((-0.1,-30)\) and \((0.1, -15)\) on the graph to complete the graph representation.
Key Concepts
Slope and Y-InterceptGraphing Linear EquationsCoordinate Plane
Slope and Y-Intercept
In the world of linear functions, the slope and y-intercept play crucial roles. The slope is a measure of the steepness of a line. It is represented by the letter \( m \) in the equation \( y = mx + b \). A slope of 75, for example, indicates a very steep line that rises quickly as we move from left to right on a graph.
A positive slope, like 75, implies that as \( x \) increases, so does \( y \). In contrast, a negative slope would suggest that \( y \) decreases as \( x \) increases.The y-intercept, denoted by \( b \), is the point where the line crosses the y-axis. In our example, \( b = -22.5 \), meaning when \( x = 0 \), \( y \) will be -22.5. The y-intercept is the starting point of the line on the graph.
Understanding both these elements helps in predicting how the graph of a linear equation will look without needing to plot multiple points.
A positive slope, like 75, implies that as \( x \) increases, so does \( y \). In contrast, a negative slope would suggest that \( y \) decreases as \( x \) increases.The y-intercept, denoted by \( b \), is the point where the line crosses the y-axis. In our example, \( b = -22.5 \), meaning when \( x = 0 \), \( y \) will be -22.5. The y-intercept is the starting point of the line on the graph.
Understanding both these elements helps in predicting how the graph of a linear equation will look without needing to plot multiple points.
Graphing Linear Equations
Graphing linear equations involves plotting points on a graph and connecting them to draw a line. It starts with identifying key components of the linear equation, notably the slope and y-intercept, typically expressed in the form \( y = mx + b \).
Steps to graph a linear equation include:
This process was exemplified in the original exercise, which involved finding points by substituting specific \( x \)-values into the function and subsequently plotting them.
Steps to graph a linear equation include:
- Identifying the slope and y-intercept from the equation.
- Calculating the \( y \)-values for given \( x \)-values based on the equation.
- Plotting these \( x, y \) pairs on a graph.
- Drawing a line through the points to represent the entire linear equation.
This process was exemplified in the original exercise, which involved finding points by substituting specific \( x \)-values into the function and subsequently plotting them.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It's composed of two axes that intersect at the origin (0,0). The horizontal axis is called the x-axis, and the vertical axis is the y-axis. These axes divide the plane into four quadrants.
When graphing linear equations, the coordinate plane becomes essential. Each point on the plane is designated by a pair of numbers, such as \((-0.1, -30)\) or \((0.1, -15)\). The first number tells us how far left or right the point is from the origin (along the x-axis), and the second number tells us how far up or down it is (along the y-axis).To plot a point:
When graphing linear equations, the coordinate plane becomes essential. Each point on the plane is designated by a pair of numbers, such as \((-0.1, -30)\) or \((0.1, -15)\). The first number tells us how far left or right the point is from the origin (along the x-axis), and the second number tells us how far up or down it is (along the y-axis).To plot a point:
- Start from the origin.
- Move horizontally to the x-coordinate value.
- Then, move vertically to the y-coordinate value.
Other exercises in this chapter
Problem 102
Graph the linear function \(f\) on a domain of [-10,10] for the function whose slope is \(\frac{1}{8}\) and \(y\) -intercept is \(\frac{31}{16}\). Label the poi
View solution Problem 103
Graph the linear function \(f\) on a domain of [-0.1,0.1] for the function whose slope is 75 and \(y\) -intercept is -22.5. Label the points for the input value
View solution Problem 104
Graph the linear function \(f\) where \(f(x)=a x+b\) on the same set of axes on a domain of [-4,4] for the following values of \(a\) and \(b\). a. \(a=2 ; b=3\)
View solution Problem 104
L. Graph the linear function \(f\) where \(f(x)=a x+b\) on the same set of axes on a domain of \([-4,4]\) for the following values of \(a\) and \(b .\) $$ a=2 ;
View solution