Finding the Axis of Symmetry in the Quadratic Equation f(x) = 3x^2 + 2

Q: What does b=0 mean in this equation?

When b=0, there's no linear term (no x term). This means the parabola is perfectly centered on the y-axis, so the axis of symmetry is $ x = 0 $.

Q: How can I see the axis of symmetry on the graph?

The axis of symmetry is the vertical line that cuts the parabola into two mirror-image halves. For $ f(x) = 3x^2 + 2 $, this line is the y-axis itself.

Q: Does the coefficient 3 affect the axis of symmetry?

No! The coefficient a=3 makes the parabola narrower and steeper, but doesn't move it left or right. Only the b coefficient determines horizontal position.

Q: What if I can't remember the axis formula?

Think about it logically: the axis passes through the vertex. For $ f(x) = 3x^2 + 2 $, the lowest point occurs when the squared term equals zero, so x=0.

Q: Why is the axis x=0 and not y=0?

The axis of symmetry is always a vertical line, so it's written as x = [number]. It shows which x-value the parabola is centered around, not a y-value.

Axis of Symmetry with Standard Form

A quadratic equation is graphed below.

What is the axis of symmetry for the graph $f(x)=3x^2+2$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's find the axis of symmetry for this function.

00:13 The axis of symmetry is the X value at the vertex, the top or bottom point of the curve.

00:19 Imagine folding the parabola in half—this line makes both sides match perfectly!

00:27 First, let's examine the function's coefficients.

00:30 We'll use a formula to find the vertex. It's a special point!

00:35 Plug in the numbers, solve for X, and that's how we find the axis!

00:40 Great job! You've solved the problem.

Step-by-step written solution

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Understand the problem

A quadratic equation is graphed below.

What is the axis of symmetry for the graph $f(x)=3x^2+2$ ?

Step-by-step solution

To determine the axis of symmetry for the function $f(x) = 3x^2 + 2$ , we follow these steps:

Identify the coefficients from the function. Here, $a = 3$ and $b = 0$ . The constant term $c = 2$ does not affect the axis of symmetry.
Use the axis of symmetry formula for a quadratic function: $x = -\frac{b}{2a}$ .
Substitute the values for $b$ and $a$ into the formula: $x = -\frac{0}{2 \times 3} = 0$ .

This calculation shows that the axis of symmetry for the graph of the quadratic function is $x = 0$ .

Thus, the solution to the problem is that the axis of symmetry is $x = 0$ .

Final Answer

$x=0$

Key Points to Remember

Essential concepts to master this topic

Formula: Use $x = -\frac{b}{2a}$ for any quadratic function
Technique: Identify coefficients: a=3, b=0, so $x = -\frac{0}{2(3)} = 0$
Check: Graph is symmetric about x=0, vertex at (0,2) confirms answer ✓

Common Mistakes

Avoid these frequent errors

Using the constant term c in the formula
Don't include c=2 in the axis formula = $x = -\frac{2}{6}$ which is wrong! The constant term only shifts the parabola up or down, not left or right. Always use only coefficients a and b in $x = -\frac{b}{2a}$ .

Practice Quiz

Test your knowledge with interactive questions

Given the expression of the quadratic function

Finding the symmetry point of the function

$$ f(x)=-5x^2+10 $$

FAQ

Everything you need to know about this question

What does b=0 mean in this equation?

When b=0, there's no linear term (no x term). This means the parabola is perfectly centered on the y-axis, so the axis of symmetry is $x = 0$ .

How can I see the axis of symmetry on the graph?

The axis of symmetry is the vertical line that cuts the parabola into two mirror-image halves. For $f(x) = 3x^2 + 2$ , this line is the y-axis itself.

Does the coefficient 3 affect the axis of symmetry?

No! The coefficient a=3 makes the parabola narrower and steeper, but doesn't move it left or right. Only the b coefficient determines horizontal position.

What if I can't remember the axis formula?

Think about it logically: the axis passes through the vertex. For $f(x) = 3x^2 + 2$ , the lowest point occurs when the squared term equals zero, so x=0.

Why is the axis x=0 and not y=0?

The axis of symmetry is always a vertical line, so it's written as x = [number]. It shows which x-value the parabola is centered around, not a y-value.

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