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You likely already know how to find the
$y$
-intercept
of a graph by looking at the image: just look for the point(s) where
the graph intersects the
**
$y$
-axis**. That method is great for graphs that include points such as
$\left(0,4\right)$
or
$\left(0,-3\right)$
, but what if the
function isn't
graphed for you or has a
$y$
-intercept that doesn't cleanly fit the scale?

Luckily, finding the $y$ -intercept of such functions is still a fairly straightforward process. If we have the equation for a function, we can find its $y$ -intercept(s) by setting $x$ to $0$ and solving for $y$ . In this article, we'll explore how to do this for several different types of functions. Let's get started!

The easiest way to go about this is to look at a practice problem, so let's find the $y$ -intercept of the following parabola:

Admittedly, we can just look at the graph and see that the $y$ -intercept is $\left(0,2\right)$ $\left(0,4\right)$ , so we'll use that to check our work. If we didn't already have that information, we could find the $y$ -intercept by taking the function $y={x}^{2}-3x+2$ and substituting 0 for $x$ , solving for $y$ . So:

$y={x}^{2}-3x+2$

becomes

${y}_{i}={\left(0\right)}^{2}-{3\left(0\right)}_{}+2$

${y}_{i}=0-0+2=2$

${y}_{i}=2$

Working out algebraic expressions with lots of zeros is generally pretty easy since any number multiplied by zero is zero, and we've determined that $y=2$ . We should always express $y$ -intercepts in the form of an ordered pair, so the answer is $\left(0,2\right)$ .

a. What is the $y$ -intercept of the following function: $f\left(x\right)=3x$

We can find the $y$ -intercept of this equation by setting $x$ to zero and solving for $y$ (or $f\left(x\right)$ in this case). That gives us:

$f\left(0\right)=3\left(0\right)$

$f\left(0\right)=0$

We want to express our answer as an ordered pair, so the $y$ -intercept is $\left(0,0\right)$ .

b. Determine the $y$ -intercept of the following function that passes through the point $\left(-4,6\right)$ :

$y={x}^{2}+3x+c$

This problem has two parts. First, we need to use the coordinates
given to us to figure out what **c** is and complete the
function. Then, we need to sub 0 for x in that function and solve
for y. Let's start by solving for c setting x to -4 and y to 6:

$6={\left(-4\right)}^{2}+3{\left(-4\right)}^{}+c$

$6=16-12+c$

$-10=-12+c$

$2=c$

That means our function is $y={x}^{2}+3x+2$ . Now, we can replace $x$ with 0 and solve for $y$ to determine the $y$ -intercept:

${y}_{i}={\left(0\right)}^{2}+{3\left(0\right)}_{}+2$

${y}_{i}=0+0+2$

${y}_{i}=2$

Since we want to express our answer as an ordered pair, the $y$ -intercept is $\left(0,2\right)$ .

c. Find the $y$ -intercept of $f\left(x\right)=3{x}^{2}+2x+7$

We can solve this problem by swapping x for zero and solving it. Our equation looks like:

$f\left(0\right)=3{\left(0\right)}^{2}+2{\left(0\right)}_{}+7$

$f\left(0\right)=0+0+7$

$f\left(0\right)=7$

Our ordered pair for the $y$ -intercept is $\left(0,7\right)$ . By now, you may be noticing a pattern with these practice problems. When solving the equations, all of the terms with an $x$ end up being zeros, meaning that the constant doubles as the $y$ -intercept. We can use this as a shortcut provided all of the other terms involve multiplying by $x$ .

Functions: Graphs and Intersections

College Algebra Diagnostic Tests

Understanding $y$ -intercepts is essential to understanding what functions look like once graphed, especially if we don't intend to plot any points. The underlying math is relatively simple, but students could still mistake the $y$ -intercept for the $x$ -intercept or make an error because they tried to rush. Fortunately, a math tutor can provide the personalized support your student needs to feel more comfortable working with functions. Contact the Educational Directors at Varsity Tutors today for more information, including a personalized quote.

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