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Graphing linear equations Interpreting graphs Plot points on a graphHere 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.

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, q**uadratic 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=18Now 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.

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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DOWNLOAD FREEIn 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}.**

Find the y intercept of the line y=2x-5.

**Substitute \textbf{x} = \bf{0} into the equation of the line.**

2**Solve 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).

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).

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).

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).

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).

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).

- 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.

**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).

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}

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.

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.

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.

- Rate of change
- Systems of equations
- Number patterns
- Geometry
- Angles
- Angles in parallel lines

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