Find the Axis of Symmetry for the Quadratic Function: f(x) = x² + 4x

Q: Why is there a negative sign in the axis of symmetry formula?

The negative sign comes from completing the square! When we rewrite $ x^2 + 4x $ as $ (x + 2)^2 - 4 $, the vertex is at $ x = -2 $, which matches $ x = -\frac{4}{2(1)} $.

Q: What if the coefficient 'a' is negative?

The formula $ x = -\frac{b}{2a} $ still works! Just be careful with signs. For example, if $ f(x) = -x^2 + 4x $, then $ a = -1 $ and $ x = -\frac{4}{2(-1)} = 2 $.

Q: Can I find the axis of symmetry without the formula?

Yes! You can complete the square or find where the derivative equals zero. But the formula $ x = -\frac{b}{2a} $ is the fastest method for standard form quadratics.

Q: How do I know if my axis of symmetry is correct?

Check that points equal distances from the axis have the same y-value. For $ x = -2 $: $ f(-1) = f(-3) = -3 $ and $ f(0) = f(-4) = 0 $ ✓

Q: What if there's no 'c' term like in this problem?

That's perfectly fine! When $ f(x) = x^2 + 4x $, we have $ c = 0 $. The axis of symmetry formula only uses 'a' and 'b', so missing 'c' doesn't matter.

Quadratic Functions with Formula Application

A quadratic function is graphed below.

What is the axis of symmetry for the graph $f(x)=x^2+4x$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's find the axis of symmetry for this function.

00:11 The axis of symmetry is the X value at the vertex point.

00:16 Imagine folding the parabola in half at this point. Both sides would match exactly.

00:22 First, we'll examine the function's coefficients.

00:26 Let's use the vertex formula to find this point.

00:30 We'll plug in the given values and calculate X for the vertex.

00:35 By finding X, we get our axis of symmetry.

00:47 And there you go, that's how you solve it!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A quadratic function is graphed below.

What is the axis of symmetry for the graph $f(x)=x^2+4x$ ?

Step-by-step solution

To solve this problem, we'll determine the axis of symmetry using the appropriate formula:

Step 1: Identify the coefficients $a$ , $b$ , and $c$
Step 2: Use the axis of symmetry formula
Step 3: Substitute the values and solve

Now, let's work through each step:

Step 1: The quadratic function is $f(x) = x^2 + 4x$ . Here, $a = 1$ , $b = 4$ , and $c = 0$ .

Step 2: Use the axis of symmetry formula $x = -\frac{b}{2a}$ .

Step 3: Substitute the values: $x = -\frac{4}{2 \times 1} = -\frac{4}{2} = -2$

Therefore, the axis of symmetry for the graph is $x = -2$ .

Final Answer

$x=-2$

Key Points to Remember

Essential concepts to master this topic

Formula: Axis of symmetry for $ax^2 + bx + c$ is $x = -\frac{b}{2a}$
Technique: From $f(x) = x^2 + 4x$ , use $a = 1, b = 4$ to get $x = -\frac{4}{2(1)} = -2$
Check: Vertex at $x = -2$ gives $f(-2) = (-2)^2 + 4(-2) = -4$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong sign in the formula
Don't forget the negative sign and write $x = \frac{b}{2a}$ = $x = \frac{4}{2} = 2$ ! This gives the wrong direction for the axis. Always remember the formula is $x = -\frac{b}{2a}$ with a negative sign.

Practice Quiz

Test your knowledge with interactive questions

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

$$ f(x)=-3x^2+3 $$

FAQ

Everything you need to know about this question

Why is there a negative sign in the axis of symmetry formula?

The negative sign comes from completing the square! When we rewrite $x^2 + 4x$ as $(x + 2)^2 - 4$ , the vertex is at $x = -2$ , which matches $x = -\frac{4}{2(1)}$ .

What if the coefficient 'a' is negative?

The formula $x = -\frac{b}{2a}$ still works! Just be careful with signs. For example, if $f(x) = -x^2 + 4x$ , then $a = -1$ and $x = -\frac{4}{2(-1)} = 2$ .

Can I find the axis of symmetry without the formula?

Yes! You can complete the square or find where the derivative equals zero. But the formula $x = -\frac{b}{2a}$ is the fastest method for standard form quadratics.

How do I know if my axis of symmetry is correct?

Check that points equal distances from the axis have the same y-value. For $x = -2$ : $f(-1) = f(-3) = -3$ and $f(0) = f(-4) = 0$ ✓

What if there's no 'c' term like in this problem?

That's perfectly fine! When $f(x) = x^2 + 4x$ , we have $c = 0$ . The axis of symmetry formula only uses 'a' and 'b', so missing 'c' doesn't matter.

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