**Y-Intercept Overview**

**Definition:**

The *y***-intercept** is the point where a graph crosses the *y*-axis. In other words, it is the value of *y* when \(x=0\).

**Y-Intercept Sample Questions and FAQs**

**How to find the ***y*-intercept:

*y*-intercept:

There is more than one way to find the *y*-intercept, depending on your starting information. Below are three ways to identify the *y*-intercept on a graph, in a table, or with an equation:

**Find the***y*-intercept of a linear function using the slope and a given point

First, identify the slope and a point on the graph. Once this is done, write a linear equation in slope-intercept form (*y = mx + b*). Using the given point (*x, y*) and the slope *m*, rewrite the equation by substituting the appropriate values for *x*, *y*, and *m*. Given this information, solve the equation for *b* to identify the *y*-intercept.

Example: Consider a graph containing the point (-2, 5) where the slope is 3.

y = | mx + b | Start by writing a linear equation in slope-intercept form. |

5 = | (3)(-2) + b | Next, replace variables y, m, and x with their corresponding values. |

5 = | -6 + b | From here, solve the one-step equation for b. |

11 = | b | b = 11, so the y-intercept is 11. |

**Find the***y*-intercept of a linear function using two points from a table or graph

Using a table or a graph, identify two points shown. First, record the coordinates (*x, y*) for each point. Using this information, find the rise and run to identify the **slope**. Calculate the rise by finding the difference in the *y*-coordinates of the two points. Calculate the run by finding the difference in the *x*-coordinates of these two points. Divide the difference in *y*-coordinates by the difference in *x*-coordinates to find the slope.

Once the slope has been identified, write a linear equation in **slope-intercept form** (*y = mx + b*). Using one set of coordinates (*x, y*) and the slope *m*, rewrite the equation by substituting the appropriate values for *x*, *y*, and *m*. Then, solve the equation for *b* to identify the *y*-intercept.

*Y***-intercept Example**

Consider a graph containing the points (3, 6) and (-1, -2). Find the *y*-intercept.

\(\frac{rise}{run}=\) | \(\frac{6-(-2)}{3-(-1)}=\frac{8}{4}=2\) | Start by calculating the rise and run to find the slope. The difference in y-coordinates is 8 and the difference in x-coordinates is 4. 8 ÷ 4 = 2, so the slope = 2. |

y = | mx + b | Next, write an equation in slope-intercept form. |

6 = | (2)(3) + b | Replace variables y, m, and x with the corresponding values. Choose one point given to substitute for x and y. |

6 = | 6 + b | Then, solve the one-step equation for b. |

0 = | b | b = 0, so the y-intercept is 0. |

**Find the***y*-intercept of a linear function using an equation

If you already have the equation of the line, solve algebraically to find the *y*-intercept. Since the *y*-intercept always has a corresponding *x*-value of 0, replace *x* with 0 in the equation and solve.

Example: Find the *y*-intercept of the line \(3x+(-2y)=12\)

3(0) + (-2y) | = 12 | First, rewrite the equation by substituting 0 for x. |

0 + -2y | = 12 | Next, solve the one-step equation for y. |

y | = -6 | y = -6, so the y-intercept is -6. |

**Finding the ***y*-intercept in a quadratic function

*y*-intercept in a quadratic function

In a quadratic function, the *y*-intercept is the point at which the parabola crosses the *y*-axis. In the graph shown, the *y*-intercept is -3.

The standard form of a **quadratic equation** is written as \(y=ax^2+bx+c\), where *x* and *y* are variables and *a*, *b*, and *c* are known constants. To find the *y*-intercept from a quadratic equation, substitute 0 as the value for *x* and solve. The *y*-intercept is always equal to the value of *c* in the equation.

Example: Find the *y*-intercept in the quadratic equation \(y=2x^2+3x+4\).

y = | 2(0)2 + 3(0) + 4 | First, rewrite the equation by substituting 0 for x. |

y = | 0 + 0 + 4 | Next, solve the one-step equation for y. |

y = | 4 | y = 4, so the y-intercept is 4. |

## Frequently Asked Questions

#### Q

### How do you find the \(y\)-intercept?

#### A

There is more than one way to find the \(y\)-intercept, depending on your starting information. If the linear equation is given, solve algebraically to find the \(y\)-intercept. Since the \(y\)-intercept always has a corresponding \(x\)-value of \(0\), replace \(x\) with \(0\) in the equation and solve for \(y\).

On a graph, the \(y\)-intercept can be found by finding the value of \(y\) when \(x=0\). This is the point at which the graph crosses through the \(y\)-axis.

#### Q

### What is the \(y\)-intercept of an equation?

#### A

When the equation of a line is written in **slope-intercept form** \((y=mx+b)\), the \(y\)-intercept is the constant, which is represented by the variable \(b\). For example, in the linear equation \(y=4x-5\), the \(y\)-intercept is \(-5\).

#### Q

### Where is the \(y\)-intercept on a graph?

#### A

The \(y\)-intercept is the point where the graph of a line crosses the \(y\)-axis. In the coordinate plane shown, the \(y\)-intercept is \(4\) because the graph passes through \(4\) on the \(y\)-axis.

#### Q

### Why is the \(y\)-intercept important?

#### A

The \(y\)-intercept is important because it tells the value of \(y\) when \(x=0\). It provides a starting point for a linear function.

#### Q

### How do I find slope and \(y\)-intercept?

#### A

On a graph, the \(y\)-intercept is the point where the line intersects the \(y\)-axis. The corresponding \(x\)-coordinate is always \(0\). The **slope** is found by calculating rise over run. This is done by finding the difference in the \(y\)-coordinates and \(x\)-coordinates and dividing these differences.

When a linear equation is written in **slope-intercept form** \((y=mx+b)\), the slope is represented by the variable \(m\). It is the coefficient to \(x\) in the equation. The \(y\)-intercept is the constant, represented by the variable \(b\).

#### Q

### Is \(b\) the \(y\)-intercept?

#### A

When a linear equation is written in **slope-intercept form** \((y=mx+b)\), the \(y\)-intercept is represented by the constant variable \(b\). For example, in the equation \(y=6x+8\), the variable \(b\) corresponds with \(8\). This is the \(y\)-intercept.

#### Q

### What does the \(y\)-intercept mean in real life?

#### A

The \(y\)-intercept is the \(y\)-value that corresponds to \(x\) when \(x=0\). In real life, this often refers to the starting point when something is being measured.

For instance, consider population change in the United States. In this scenario, the \(x\)-values could represent time, measured in years. The \(y\)-values could represent the population, measured in millions of people. When \(x=0\), this value represents the starting year for measuring population change. The corresponding \(y\)-value represents the size of the population in the starting year. This value is the \(y\)-intercept.

## Practice Questions

**Question #1:**

The function \(y=\frac{1}{2}x+3\) is graphed below. Use the graph to identify the y-intercept.

y-intercept = 3

y-intercept = 2

y-intercept = 4

y-intercept = \(\frac{1}{2}\)

**Answer:**

The correct answer is A. The y-intercept is the point where the graph crosses the y-axis. When studying the graph above, notice that the line crosses the y-axis at (0, 3), so 3 is the y-intercept.

**Question #2:**

Which variable represents the y-intercept for a quadratic equation in standard form:

y = ax^{2} + bx + c

*a*

*b*

*c*

*y*

**Answer:**

The correct answer is C. In a quadratic equation, the variable c represents the y-intercept. This is the point where the graph intersects the y-axis.

**Question #3:**

Without graphing, identify the y-intercept for the function \(y=-4x+\frac{1}{2}\)

2

4

**Answer:**

The correct answer is D. When a function is in slope-intercept form, the y -intercept can quickly be identified because it is represented by the variable b. In this example, b is \(\frac{1}{2}\), therefore the y-intercept is \(\frac{1}{2}\).

**Question #4:**

The quadratic equation y = -3x^{2} – 3x + 1 is graphed below. What is the y-intercept?

y-intercept = 0

y-intercept = 1

y-intercept = 0.25

y-intercept = -1.5

**Answer:**

The correct answer is B. The y-intercept is the point where the graph crosses the y-axis. When studying the graph above, notice that the line crosses the y-axis at 1, so 1 is the y-intercept.

**Question #5:**

Which equation is represented by the graph below? Use your understanding of y -intercept to determine your answer.

**Answer:**

The correct answer is B. The quadratic equation \(y=2x^2-3x+4\) is written in standard form. This makes it easier to identify the y-intercept, because c always represents the y-intercept. The graph shows a quadratic equation that intersects the y-axis at 4, and the only equation where c is equal to 4 is Choice B.