What Is the `y`-intercept

• Introduction
• What Is the `y`-intercept?
• How to Find `y`-intercept?
• Real Life Examples of `y`-intercept
• The `y`-Intercept Formula
• What Is the `x`-Intercept? 
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Introduction

Understanding the intercepts is important, as they provide important information about the behavior of the graph of function. The points where the graph of a function meets one of the axes on a coordinate plane, the `x`-axis or the `y`-axis are called intercepts. The point where the graph of a function crosses the `y`-axis is called the `y`-intercept, and the point(s) where it crosses the `x`-axis is called the `x`-intercept(s).

What Is the `y`-intercept?

`y`-intercept is the point where the graph of a function touches or crosses the `y`-axis. At this point, the value of `x` is zero. We use this fact to find the `y`-intercept algebraically from a given function. To find the `y`-intercept, we evaluate the function by substituting `x = 0`.

How to Find `y`-intercept?

How Do You Find The `y`-intercept From a Graph?

To find the `y`-intercept from the graph, we look for the point where the graph crosses the `y`-axis.

How Do You Find the `y`-Intercept From an Equation?

At `y`-intercept of a graph, the value of `x` is zero. If we have an equation representing the function, we can find the `y`-intercept by substituting `x` equal to `0` in the equation and solving for `y`.

How Do You Find the `y`-Intercept From a Table?

Let’s understand this using the following example: 

When looking at points on a line, if you see that every time the \( x \)-value changes by the same amount, the \( y \)-value also changes by the same amount, which is called a constant change. This tells us where a specific \( y \)-value occurs on the line, based on a particular \( x \)-value.

From the points `(-2, -6)` and `(1, 9)`, we see that every time \( x \) increases by \( 3 \), the \( y \) value increases by \( 15\). 

Let’s start at `(3,19)`, and get the table backward. So, if we go back \( 3 \) step from \( x = 3 \) to \( x = 0 \), we estimate that the \( y \) value will decrease by `15` units. So, `19-15 = 4`. So,  \( y \)-intercept is \(0, 4 \).

Real Life Examples of `y`-intercept

The \( y \)-intercept is a point on a graph where the output variable (like the height or the temperature) is any value, while the input variable (like time or distance) is zero. It's where the graph crosses the output variable axis when the input variable is set to zero.

Some `y` intercept examples:

`1`. Cost Function: Let’s take an example of running a lemonade stand. The cost function shows how much money is spent (output) based on the number of lemonades made (input). The \( y \)-intercept of this cost function would be where no lemonades are made.

`2`. Height Example: Suppose a height function models the height of a ball thrown into the air over time. The height (output) depends on how much time has passed (input). Before your first throw the ball (time `= 0`), the height at that moment is your \( y \)-intercept.

The `y`-Intercept Formula

What Is the `x`-Intercept? 

The point where the graph of a function crosses the `x`-axis is called the `x`-intercept. At this point, the value of `y` is zero. To find the `x`-intercept algebraically, set the function equal to zero and solve for `x`.

Solved Examples

Example `1`: A company's total cost \( C \) for producing \( x \) units of a product is given by the equation \( C(x) = 2x^2 - 5x + 10 \). Find  \( y \)-intercept of this equation. What does the \( y \)-intercept of this equation represent? 

Solution:

To find the \( y \)-intercept, we set \( x = 0 \) in the equation:

\( C = 2(0)^2 - 5(0) + 10 = 10\)

So,  \( y \)-intercept is `(0,10)`, which represents the initial cost is `$10`. 

Example `2`: Given the table below, determine the \( y \)-intercept of the linear equation represented by the data.

Solution:

From the points `(-4, 8)` and `(-2, 5)`, we can understand that for every `2` unit increase in \( x \), `y` decreases by `3` units.

Let’s start at `(-2,5)`, and get the table forward. So, if we go \( 2 \) units forward from \( x = -2 \) to \( x = 0 \), we estimate that the \( y \) value will decreases by `3` units. So, `5-3 = 2`. So,  \( y \)-intercept is `(0, 2)`.

Example `3`: Find the \( y \)-intercept of the linear equation \( y = 2x + 4 \).

Solution:

To get the `y` intercept, set `x=0` in the given equation and solve for `y`. 

\( y = 2(0) + 4 = 4 \)

So, `y` intercept of the equation is `(0,4)`.

Example `4`: Determine the \( y \)-intercept of the cubic equation \( f(x) = x^3 - 3x^2 + 2x - 5 \).

Solution:  

To get the `y` intercept of this cubic function, set `x=0` and evaluate the value of the function at that point.  

\( f(x) = x^3 - 3x^2 + 2x - 5 \)

 \(f(0) = (0)^3 - 3(0)^2 + 2(0) - 5 = -5\)

So, `y` intercept would be `(0,-5)`.

Example `5`: What is the `y` intercept of the given graph?

Solution:

The `y` intercept is the point where the graph of the function crosses the `y` axes. Let’s identify the point where the function crosses the `y` axes.

We can see that the graph passes through the `y` axes at the point `(0,3)`. So, the `y`-intercept would be `(0,3)`.

Practice Problems

Q`1`. The height \( h \) of a ball thrown upward can be modeled by the equation \( h = -5t^2 + 10t + 15 \), where \( t \) is the time in seconds. What does the \( y \)-intercept of this equation represent?

1. The initial height of the ball
2. The time when the ball hits the ground
3. The maximum height the ball reaches
4. The time it takes for the ball to reach its highest point

Answer: a

Q`2`. Given the table below, determine the \( y \)-intercept of the linear equation represented by the data. 

1. `-2`
2. `4`
3. `2`
4. `10`

Answer: c

Q`3`. Find the \( y \)-intercept of the equation \( y = -3x + 9 \).

1. `(-3, 0)`
2. `(0, -3)`
3. `(9, 0)`
4. `(0, 9)`

Answer: d

Q`4`. Determine the \( y \)-intercept of the line given by the equation \( 2y = 4x - 8 \).

1. `(4, 0)`
2. `(0, -4)`
3. `(0, -2)`
4. `(0, 4)`

Answer: b

Q`5`. What is the \( y \)-intercept of this graph?

1. `(2, 0)`
2. `(4, 0)`
3. `(0, 2)`
4. `(0, 4)`

Answer: d

Frequently Asked Questions

Q`1`. What is the `y`-intercept in a linear equation?

Answer: The `y`-intercept is the point where a line intersects the `y`-axis on a coordinate plane. It represents the value of output variable, when input variable is set to `0` in a function.

Q`2`. How do you find the `y`-intercept from an equation?

Answer: To find the `y` intercept when an equation is given, set the input variable equal to zero, and solve for the output variable.

Q`3`. Why is the `y`-intercept important in linear equations?

Answer: The `y`-intercept provides crucial information about the starting value or initial condition of a linear relationship. It helps in understanding the behavior of the line. It also help in graphing the function.

Q`4`. Can a line have a y-intercept but no x-intercept?

Answer: Yes, it is possible that a line can have a `y`-intercept but no `x`-intercept. When the line is horizontal, and its equation is in the form `y =` constant.

Q`5`. How does the `y`-intercept relate to real-world scenarios?

Answer: In real-world applications, the `y`-intercept often represents the initial value of a quantity. For example, in a distance-time graph, the `y`-intercept can represent the starting position or height of an object at time zero.