Find X-Intercepts: Solving y = 16 - x² When y Equals Zero

Q: What exactly are x-intercepts?

X-intercepts are points where the graph crosses the x-axis. At these points, the y-value is always zero, so you set the equation equal to 0 to find them.

Q: Why do I get two answers when solving x² = 16?

Because both positive and negative numbers give the same result when squared! Since $ 4^2 = 16 $ and $ (-4)^2 = 16 $, you get two x-intercepts.

Q: What if I get a negative number under the square root?

If you get something like $ x^2 = -9 $, there are no real x-intercepts because you can't take the square root of a negative number in real numbers.

Q: Can a parabola have more than two x-intercepts?

No! A parabola (quadratic function) can have at most 2 x-intercepts. It might have 2, 1, or 0 x-intercepts, but never more than 2.

Quadratic Functions with X-Intercept Solutions

Determine the points of intersection of the function

$y=16-x^2$

With the X

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find where the function crosses the X-axis.

00:10 At these points, Y is always zero.

00:14 So, let's plug in Y equals zero into our equation.

00:19 Next, we will solve to find the value of X.

00:27 Now, let's isolate X by extracting the root.

00:32 Remember, finding a root gives us two solutions: positive and negative.

00:37 These are the possible X values.

00:41 Since Y is zero, as we set it from the start.

00:52 Great job! That's how we solve for these intersection points.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Determine the points of intersection of the function

$y=16-x^2$

With the X

Step-by-step solution

To determine the points of intersection of the function $y = 16 - x^2$ with the x-axis, we follow these steps:

Step 1: Set the equation equal to zero since the x-axis corresponds to $y = 0$ .
$y = 16 - x^2 = 0$
Step 2: Rearrange the equation to isolate $x^2$ .
Adding $x^2$ to both sides gives: $x^2 = 16$ .
Step 3: Solve the equation $x^2 = 16$ for $x$ by taking the square root of both sides.
$x = \pm \sqrt{16}$ results in $x = \pm 4$ .
Step 4: Determine the intersection points. For $x = 4$ , the point of intersection is $(4, 0)$ , and for $x = -4$ , the point of intersection is $(-4, 0)$ .

Therefore, the points of intersection of the parabola with the x-axis are $(-4,0)$ and $(4,0)$ .

This corresponds to the answer choice: $(-4,0),(4,0)$ .

Final Answer

$(-4,0),(4,0)$

Key Points to Remember

Essential concepts to master this topic

Rule: Set function equal to zero to find x-intercepts
Technique: Solve $x^2 = 16$ gives $x = \pm 4$
Check: Substitute back: $y = 16 - (4)^2 = 0$ and $y = 16 - (-4)^2 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative solution when taking square roots
Don't write just x = 4 when solving $x^2 = 16$ = missing half the answer! Square roots always give both positive and negative solutions. Always write $x = \pm \sqrt{16} = \pm 4$ to find both x-intercepts.

Practice Quiz

Test your knowledge with interactive questions

Which chart represents the function $$ y=x^2-9 $$ ?

FAQ

Everything you need to know about this question

What exactly are x-intercepts?

X-intercepts are points where the graph crosses the x-axis. At these points, the y-value is always zero, so you set the equation equal to 0 to find them.

Why do I get two answers when solving x² = 16?

Because both positive and negative numbers give the same result when squared! Since $4^2 = 16$ and $(-4)^2 = 16$ , you get two x-intercepts.

How do I write the final answer as coordinate points?

X-intercepts are always on the x-axis where y = 0. So if your x-values are 4 and -4, write them as coordinate points: $(4, 0)$ and $(-4, 0)$ .

What if I get a negative number under the square root?

If you get something like $x^2 = -9$ , there are no real x-intercepts because you can't take the square root of a negative number in real numbers.

Can a parabola have more than two x-intercepts?

No! A parabola (quadratic function) can have at most 2 x-intercepts. It might have 2, 1, or 0 x-intercepts, but never more than 2.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Parabola Families questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime