Here you will learn about how to find the y intercept from a straight line graph, including straight lines in the slope intercept form, y=mx+b and standard form, ax+by=c.
Students will first learn about how to find the y intercept in 8 th grade math with their work with linear functions.
What is the y intercept and how do you find the y -intercept?
Finding the y intercept of a straight line is an important skill used to solve algebraic and real-life problems involving straight line graphs.
The intercepts of a graph are where the graph crosses the coordinate axes.
You can find the y intercepts of graphs of all types of functions including straight lines, quadratic functions, cubic functions and others. In this case, the y intercept is the point where the function crosses or intercepts the y -axis.
To find the y intercept(s) of a line:
Substitute x=0 into the equation of the function and evaluate for y.
To find the x intercept(s) of a line:
Substitute y=0 into the equation of the function and evaluate for x.
Note, this is a useful strategy to draw the equation of a straight line.
Step-by-step guide: Graphing Linear Equations
Example
The y intercept of a straight line is the value of y when the x -coordinate is zero.
In this real life example, the y intercept represents the starting value or fixed price, with the slope or gradient representing the unit rate or rate of change.
Let’s look at some examples.
A straight line has the equation, in standard form, 4x-3y=18. Find the y intercept of the line.
To find the y intercept, substitute x=0 into the equation.
4(0)-3y=18
Now solve the equation to find the corresponding y value.
\begin{aligned}-3y&=18 \\\\y&=-6 \end{aligned}
The y intercept is -6 and has coordinate (0,-6).
Note: If any function is of the form y=f(x)+\text{constant}, the constant is the y intercept. This is because the y -intercept is always when the x -value is 0, so when substituting 0 for x the constant is going to be the y -value.
See examples of different types of polynomials below.
What is the y intercept and how do you find the y-intercept?
Common Core State Standards
How does this relate to high school math?
- High School: Functions (HS.IF.C.7a)
Graph linear and quadratic functions and show intercepts, maxima, and minima.
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How to find the y intercept
In order to find the y intercept, you need to:
- Substitute \textbf{x} = \bf{0} into the equation of the line.
- Solve the equation for \textbf{y}.
How to find y intercept examples
Example 1: finding the y intercept of a line in the form y = mx + b
Find the y intercept of the line y=2x-5.
- Substitute \textbf{x} = \bf{0} into the equation of the line.
y=2(0)-5
2Solve the equation for \textbf{y}.
This equation gives y=-5.
The y intercept is -5.
It has coordinates (0,-5).
Note: This is the special case where the equation is in the form y = [function of x ] + \; c (constant). In this case c=-5 so the y intercept is (0,-5).
Example 2: finding the y intercept of a line in the form y = mx + b
Find the y intercept of the line y=\cfrac{1}{2}x+3
Substitute \textbf{x} = \bf{0} into the equation of the line.
y=\cfrac{1}{2}(0)+3
Solve the equation for \textbf{y}.
This equation gives y=3.
The y intercept is 3.
It has coordinates (0,3).
Example 3: finding the y intercept of a line in the form y = mx + b
Find the y intercept of the line y=9x-14.
Substitute \textbf{x} = \bf{0} into the equation of the line.
y=9(0)-14
Solve the equation for \textbf{y}.
This equation gives y=-14.
The y intercept is -14.
It has coordinates (0,-14).
Example 4: finding the y intercept of a line in the form ax + by = c
Find the y intercept of the line 2x+5y=20.
Substitute \textbf{x} = \bf{0} into the equation of the line.
2(0)+5y=20
Solve the equation for \textbf{y}.
\begin{aligned}5y&=20 \\\\ y&=4 \end{aligned}
The y intercept is 4.
It has coordinates (0,4).
Example 5: finding the y intercept of a line in the form ax + by = c
Find the y intercept of the line 3x-4y=24.
Substitute \textbf{x} = \bf{0} into the equation of the line.
3(0)-4y=24
Solve the equation for \textbf{y}.
\begin{aligned}-4y&=24 \\\\ y&=-6 \end{aligned}
The y intercept is -6.
It has coordinates (0,-6).
Example 6: finding the y intercept of a line in the form ax + by = c
Find the y intercept of the line 7x+9y=36.
Substitute \textbf{x} = \bf{0} into the equation of the line.
7(0)+9 y=36
Solve the equation for \textbf{y}.
\begin{aligned}9y&=36 \\\\ y&=4 \end{aligned}
The y intercept is 4.
It has coordinates (0,4).
Teaching tips for how to find y intercept
- While introducing the concept of the y -intercept, use multiple visuals where students can clearly see that the y -intercept is the point where the line crosses the y -axis.
- The use of interactive activities, such as online graphing tools or activities that require students to physically graph activities, are just as effective as worksheets with practice problems.
- For students that are struggling, break down the process of finding the y -intercept into simple steps. The use of visuals for each step can aid students in understanding the process.
Easy mistakes to make
- Substituting the wrong value for zero
A common error is to think that the y intercept is when y=0. It is important to remember that when y=0, the line will be crossing the x -axis and therefore will give the x intercept.
- Forgetting to state the coordinates when asked
If an exam question asks for the y intercept or x intercept, just the value where the line crosses the axes is appropriate. However, sometimes the question will ask for the coordinates of the y intercept. In this case the answer must be given in the form (0,b) for the y intercept, or (-\cfrac{b}{m},0) for the coordinates of the x intercept.
- Confusing the intercept with the gradient
When the line is in the form y=mx+b, a common error is to confuse the y intercept, b with the gradient or slope of a line, m. The y intercept will have coordinates (0,b) and the x intercept will have coordinates (-\cfrac{b}{m},0).
Related graphing linear equations lessons
- Graphing linear equations
- Slope intercept form
- How to find the slope of a line
- How to find midpoint
- Distance formula
- Linear interpolation
Practice how to find the y intercept questions
1. State the coordinate of the y intercept of the line y=3x-2.
(-2,0)
(0,-2)
(3,0)
(0,3)
Substitute x=0 to give
\begin{aligned}y&=3(0)-2 \\\\y&=-2 \end{aligned}
The coordinate of the y -intercept is (0,-2).
2. Find the coordinate of the y intercept of the line y=3x-6.
(0,-6)
(0,2)
(2,0)
(-6,0)
Substitute x=0 to give
\begin{aligned}y&=3(0)-6 \\y&=-6 \end{aligned}
The coordinate of the y -intercept is (0,-6).
3. The equation of a line is given as y=8-3x. Find the y intercept.
\cfrac{8}{3}
-3
-8
8
When x=0,
\begin{aligned}y&=8-3(0) \\\\y&=8\end{aligned}
4. The equation of a line is given as x-3y=9. Find the y intercept.
-3
3
-\cfrac{1}{3}
9
When x=0,
\begin{aligned}-3y&=9 \\\\y&=-3 \end{aligned}
5. The equation of a line is given as 5x-4y=10. Find the y intercept.
2
-2.5
-\cfrac{2}{5}
\cfrac{1}{2}
When x=0,
\begin{aligned}5(0)-4y&=10 \\\\-4y&=10 \\\\y&=-2.5 \end{aligned}
6. The equation of a line is given as -2x+12y=6. Find the y intercept.
\cfrac{1}{2}
\cfrac{1}{6}
-2
3
When x=0,
\begin{aligned}-2(0)+12y&=6 \\\\12y&=6 \\\\y&=\cfrac{6}{12}=\cfrac{1}{2} \end{aligned}
How to find y intercept FAQs
How do I find the \textbf{x} -intercept of a line?
The process for finding the x -intercept of a line is similar to finding the y -intercept. Instead of substituting x=0, you would substitute y=0, and solve the equation for x.
Can you find the \textbf{y} -intercept when the equation is in point-slope form?
Yes, the process for finding the y -intercept in point-slope form is similar to any other linear equation. You will substitute x=0 into the equation and then solve for y.
How do you find the slope of the line?
To find the slope of the line, you will use the formula: m=\cfrac{y_{2}-y_{1}}{x_{2}-x_{1}}, where the slope (m), is found by inserting the x -coordinate and y -coordinate from two points on a given line.
The next lessons are
- Rate of change
- Systems of equations
- Number patterns
- Geometry
- Angles
- Angles in parallel lines
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