About Graphing a Linear Equation Using Intercept Form
What is a Linear Equation?
A linear equation is defined as an equation whose variables are raised to 1. When this equation is graphed, we can see that it will always take the shape of a straight line.
A linear equation in the slope-intercept form can be expressed as:
y = mx + b, where the slope is m, and the y-intercept is b.
Example: Find the slope of the line: y = 5x + 3
The equation of the line is written in the slope-intercept form: y = mx + b, where m denotes the slope and b denotes the y-intercept. We can see that the slope of the line is 5 in the equation, y = 5x + 3.
What is the x-intercept?
The x-coordinate of the point at which a line crosses the x-axis is the x-intercept. On the x-axis, there is no y-coordinate. Hence, to find the x-intercept of an equation without drawing the graph, we just need to substitute the value of y as zero. The resulting value of x is the x-intercept.
What is the y-intercept?
The y-coordinate of the point at which a line crosses the y-axis is the y-intercept. On the y-axis, there is no x-coordinate. Hence, to find the y-intercept of an equation without drawing the graph, we just need to substitute the value of x as zero. The resulting value of y is the y-intercept.
How can we Derive a Linear Equation in the Slope-Intercept Form?
- Consider a line passing through the y-axis at the point of intersection (0, b).
- Consider a random point (x, y) on the line to find the equation of a line.
- The slope m is \(m~=~\frac{change~in~y}{change~in~x}~=~\frac{y~-~b}{x~-~0}\).
- When we solve further, we get y = mx + b.
Graphing an Equation in the Slope-Intercept Form
Example: We need to follow several steps to graph an equation in the slope-intercept form:
Identify the x-intercept by graphing y = x + 3.
Solution:
Step 1: Find the slope and the y-intercept of the given equation.
We have y = x + 3
The slope is 1 and the y-intercept is 3.
Step 2: The y-intercept is 3. So, plot (0,3).
Step 3: Find another point by using the slope and drawing a line.
\(m ~=~\frac{rise}{run}~=~11\)
So, we plot a point that is one unit right and one unit up from (0,3) which is point (1,4)
Draw a line through these points.
The line crosses the x-axis at (-3, 0). So, the x-intercept is -3.
Learn More about Slope-Intercept Form in this Video
Solved Examples on Graphing a Linear Equation with Slope Intercept Form
Example 1: Find the slope and the y-intercept of the linear equation: y = -3x + 2
Solution:
y = (-3) x + 2 Write in slope-intercept form y = mx + b
The slope is -3 and the y-intercept is 2.
Example 2: Find the slope and the y-intercept of the linear equation: \(y~-~2~=~\frac{4}{5}~x\)
Solution:
We have\(y~-~2~=~\frac{4}{5}~x\)
\(y~-~2~=~\frac{4}{5}~x~+~2\) Add 2 to each side
Now the equation is of the form y = mx + b
The slope is \(\frac{4}{5}\) and the y-intercept is 2.
Example 3: Identify the x-intercept by graphing y = 2x + 4.
Solution:
Step 1: Find the slope and the y-intercept of the given equation.
We have y = 2x + 4
The slope is 2 and the y-intercept is 4.
Step 2: The y-intercept is 4. So, plot (0,4)
Step 3: Find another point by using the slope and drawing a line.
\(m~=~\frac{rise}{run}~=~\frac{2}{1}\)
Plot the point that is one unit right and 2 units up from (0,4). We get the point as (1,6). Draw a line through these points.
The line crosses the x-axis at (-2,0). So, the x-intercept is -2.
Example 4: The equation y = 3x + 5 represents the cost (in dollars) of driving a rental car x miles. Draw the graph. Interpret the slope and the y-intercept.
Solution:
In the equation y = 3x + 5, the y-intercept is 5 and the slope is 3. So, plot the point (0,5).
\(m~=~\frac{3}{1}\)
So, plot the point 3 units up and 1 unit right from (0,5). We reach at (1,8).
Draw a line through these points. This line represents the graph of the given equation.
So, from the equation, we can see that there is an initial fee of $5 to take the rental car. The slope is 3. So, the cost per mile is $3.
Example 5: The equation y = x + 6 represents the number of bikes parked, every hour after 10 a.m. Make an equation graph. Interpret the slope and the y-intercept.
Solution:
In the equation y = x + 6, the y-intercept is 6 and the slope is 1. To graph the equation, use the slope and the y-intercept. Now plot the point (0,6)
\(m~=~\frac{1}{1}\)
Plot the point 1 unit right and 1 unit up from (0,6). We reach at (1,7)
Draw a line through these points. This line represents the graph of the given equation.
The y-intercept is 6. So, till 10 a.m., 6 bikes were already parked. The slope is 1. Hence, an extra bike was parked every hour after 10 a.m.
y = mx + d is the general formula for the slope-intercept form, where m is the slope of the line and d is the y-value of the line’s y-intercept. We can easily determine how steep a line is and where it crosses the y-axis, using the slope-intercept form of a linear equation.
The slope of horizontal lines is 0. As a result, m = 0 in the slope-intercept equation y = mx + b. The equation now is y = b, where b is the y-coordinate of the y-intercept.
A vertical line has an undefined slope, as x is constant. Hence, the change in x is zero.
The slope (m) determines how steep the graph is. The sign of the slope indicates if the line is sloping upwards or downwards.