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Magazine Chapter 1: Problem 8
Graph \(y=f(x)\) by hand by first plotting points to determine the shape of the graph. $$ f(x)=x+1 $$
Short Answer
Expert verified
The graph of \( f(x) = x + 1 \) is a straight line with a slope of 1 and y-intercept at 1.
Step by step solution
01
Understand the Function
The given function is \( f(x) = x + 1 \), which represents a linear function. The graph of a linear function forms a straight line. The slope of this line is 1 and its y-intercept is 1.
02
Create a Table of Values
To plot points, create a table of values by choosing different values for \( x \), then calculate the corresponding \( y \) values. For example, use \( x = -1, 0, 1, 2, 3 \).- \( f(-1) = -1 + 1 = 0 \)- \( f(0) = 0 + 1 = 1 \)- \( f(1) = 1 + 1 = 2 \)- \( f(2) = 2 + 1 = 3 \)- \( f(3) = 3 + 1 = 4 \)The points are \((-1, 0)\), \((0, 1)\), \((1, 2)\), \((2, 3)\), \((3, 4)\).
03
Plot the Points
On a graph, plot the points \((-1, 0)\), \((0, 1)\), \((1, 2)\), \((2, 3)\), and \((3, 4)\). Use a Cartesian plane with both x and y axes labeled, and ensure that the increments are consistent for accurate representation.
04
Draw the Line
Once the points are plotted, draw a straight line through all the points. Because this is a linear function, all points should align along a single straight line, indicating the graph of \( f(x) = x + 1 \).
05
Verify the Line
Check that the line passes through more points not originally plotted to ensure accuracy. For example, for \( x = -2 \), \( f(-2) = -2 + 1 = -1 \), and ensure this point is located on the line. Confirm additional points to ensure the line's correct position.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equations
A linear equation is a mathematical statement that describes a straight line when graphed on a coordinate plane. The general form of a linear equation is \( y = mx + b \), where:
- \( y \) is the dependent variable.
- \( m \) is the slope of the line, which indicates its steepness.
- \( x \) is the independent variable.
- \( b \) is the y-intercept, or the point where the line crosses the y-axis.
Plotting Points
Plotting points involves finding specific coordinates that satisfy the equation of a line and marking them on the coordinate plane, creating a visual representation of the equation. To do this effectively:
- Choose a few values for \( x \).
- Calculate the corresponding \( y \) values using the equation.
- Write down these pairs of \( (x, y) \) values as coordinates, like \((-1, 0)\), \((0, 1)\), etc.
- Plot these points on the graph paper.
- Ensure the points are accurately placed and in relative proportion.
Coordinate Plane
The coordinate plane is a two-dimensional surface with two axes, the x-axis, and the y-axis, intersecting at a point called the origin. This system allows us to describe the location of points in a flat space using ordered pairs. Here's how it works:
- The x-axis is horizontal and represents the input values of the function.
- The y-axis is vertical and shows the output values after applying the function.
- Coordinates are written as \((x, y)\), where \(x\) is the position on the horizontal axis, and \(y\) is the position on the vertical axis.
- This coordinate system is essential for plotting mathematical functions and visualizing their graphs.
Y-Intercept
The y-intercept of a linear equation is an important feature that provides valuable information about the graph's relation to the y-axis. It is the point where the line crosses the y-axis, giving us the value of \( y \) when \( x \) is zero.
- In the equation \( y = mx + b \), \( b \) represents the y-intercept.
- To find the y-intercept, simply evaluate the equation at \( x = 0 \).
- For example, in the function \( f(x) = x + 1 \), when \( x = 0 \), \( y = 0 + 1 = 1 \).
- This means the y-intercept is the point \((0, 1)\) on the graph.
