Question

Sketch a graph of the equation. $$ y-1=3(x+4) $$

Short answer

The equation \(y-1 = 3(x+4)\) represents a straight line with a slope of 3 and y-intercept at \(y =13\). The graph starts at point (0,13) and rises by 3 units and runs 1 unit to the right to find other points.

Step-by-step solution

Identify the slope and y-intercept

The equation given is \(y-1 = 3(x+4)\). If it's rearranged to the form \(y = mx + c\), it becomes \(y = 3x + 13\). The coefficient of x is the slope, which is 3 in this case, and the constant term is the y-intercept, which is 13.

Plot the y-intercept

The y-intercept is a point on the graph where the line crosses the y-axis. Since the y-intercept is 13, plot a point at \(y = 13\) on the y-axis.

Use the slope to find other points

The slope of the line, 3, represents the ratio of vertical change (rise) to horizontal change (run), as you move along the line. Here, rise is 3 and run is 1 (since 3 can be written as 3/1). Starting from the y-intercept, count 3 units up (since rise is positive) and 1 unit to the right (since run is positive). This is a new point that the line passes through.

Draw the line

Draw a straight line through the plotted points. This line represents the graph of the given equation \(y-1=3(x+4)\).

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing and analyzing lines efficiently. This form is expressed as
\[ y = mx + b \]
where \(m\) represents the slope and \(b\) indicates the y-intercept. The slope is a number that describes both the direction and the steepness of the line. Meanwhile, the y-intercept is the point where the line crosses the y-axis.

Transforming to Slope-Intercept Form

To sketch the graph of a linear equation, first, ensure it is in slope-intercept form. Take the equation
\( y - 1 = 3(x + 4) \)
and rearrange it to get
\( y = 3x + 13 \).
In this format, you can easily identify the slope, \(m = 3\), and the y-intercept, \(b = 13\), guiding you to graph the line quickly. It's like having a map where 'm' tells you the direction to go (up if positive, down if negative) and 'b' shows you where to start on the y-axis.

Y-Intercept

The y-intercept of a line is one of the first pieces of information you should find when graphing. It is the point where the line crosses the y-axis and is exclusively defined by its y-coordinate, as the x-coordinate is always zero at this point. In the equation \( y = 3x + 13 \), the y-intercept is \( (0, 13) \).

Plotting the Y-Intercept

To plot the y-intercept on a graph, locate the number represented by 'b' along the y-axis, and mark that point. This marks the starting point for your line. Even without knowing the slope at first, the y-intercept gives you the exact location where your line will intersect the y-axis, anchoring the line in place. For the given equation, start at 13 on the y-axis, ensuring your graph begins accurately.

Slope of a Line

The slope is a measure of the line's tilt, often referred to as 'rise over run.' It tells us how much the line rises vertically (the rise) for a given horizontal movement (the run). The slope is represented by the variable \(m\) in slope-intercept form and can be either positive, negative, zero, or undefined, depending on the direction of the line.

Calculating Slope

Given the slope-intercept form \( y = mx + b \), our slope \(m\) is the coefficient in front of \(x\). For \( y = 3x + 13 \), the slope \(m\) is 3, which means for every one unit you move to the right (the run), you move three units up (the rise).

Using Slope to Graph

When graphing, start at the y-intercept and use the slope to find other points. From \( (0, 13) \) move 3 units up and 1 unit to the right to find a new point. Then draw a straight line through these points to represent the graph of the equation. In this way, the slope directly determines the angle and direction of the line on a graph.