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Magazine Chapter 5: Problem 26
Sketch the line with the given slope and \(y\) -intercept. $$m=3,(0,1)$$
Short Answer
Expert verified
Sketch the line using points (0,1) and (1,4) with equation \( y = 3x + 1 \).
Step by step solution
01
Understand the problem
To sketch the line given the slope \( m = 3 \) and \( y \)-intercept \( (0,1) \), we need to use the slope-intercept form of a line, which is given by \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the \( y \)-intercept.
02
Identify the slope and y-intercept
From the problem, we know that the slope \( m \) is 3. This means the line rises 3 units for every 1 unit it moves to the right. The \( y \)-intercept \( b \) is 1, which means the line crosses the \( y \)-axis at the point (0, 1).
03
Write the equation of the line
Using the slope-intercept form \( y = mx + b \), substitute \( m = 3 \) and \( b = 1 \) to get the equation of the line: \[ y = 3x + 1 \]
04
Plot the y-intercept
Start by plotting the \( y \)-intercept, which is the point (0, 1), on a graph. This is where the line will cross the \( y \)-axis.
05
Use the slope to find another point
Starting from the point (0, 1), use the slope \( m = 3 \) to find another point. Since the slope is 3, move 3 units up and 1 unit to the right. This gives the point (1, 4).
06
Draw the line
Draw a straight line through the points (0, 1) and (1, 4) to complete the sketch of the line. Extend the line in both directions, and label it with the equation \( y = 3x + 1 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Linear Equations
Graphing linear equations is a key skill in understanding algebra. At the heart of this is the slope-intercept form: \( y = mx + b \), where \( m \) represents the slope and \( b \) the \( y \)-intercept. This form provides a clear framework for drawing the line on a graph.
Let's break it down:
Let's break it down:
- Start by identifying the \( y \)-intercept. This is where the line crosses the \( y \)-axis on the graph. It gives us a starting point for drawing our line.
- Next, use the slope to determine the direction and steepness of the line. A positive slope means the line ascends as it moves right, while a negative slope descends.
- Finally, connect the dots by plotting additional points using the slope and drawing a straight line through them.
y-intercept
The \( y \)-intercept is the point where a line intersects the \( y \)-axis. It is denoted by \( b \) in the slope-intercept equation \( y = mx + b \). In simpler terms, it tells you where your line begins on the vertical axis.
Here's how to make sense of it:
Here's how to make sense of it:
- On a graph, the \( y \)-intercept is always at the coordinate \((0, b)\).
- This is because when \( x = 0 \), the equation simplifies to \( y = b \).
- It's a critical point to plot first because it sets your baseline for drawing the line.
Slope of a Line
The slope of a line, represented by \( m \), informs how steep a line is. Think of it as the angle of your line in relation to the x-axis. In mathematical terms, the slope is what determines how much \( y \) changes with a change in \( x \).
Understand slope with these pointers:
Understand slope with these pointers:
- A slope of 3, for instance, indicates that for every unit you move to the right along the x-axis, the line goes up 3 units.
- It's expressed in the formula as \( m = \frac{\text{rise}}{\text{run}} \). That means, the change in \( y \) (rise) over the change in \( x \) (run).
- If the slope is positive, the line rises to the right. If negative, it falls to the right.
