Locate the Symmetry Point of the Quadratic Function f(x)=3+3x^2

Q: What exactly is a symmetry point in a quadratic function?

The symmetry point is the vertex of the parabola - the point where the graph is perfectly balanced! For $ f(x) = 3 + 3x^2 $, it's the lowest point at (0,3) where the parabola turns around.

Q: Why is b = 0 in this function?

In the standard form $ ax^2 + bx + c $, our function $ f(x) = 3 + 3x^2 $ has no x term. That means the coefficient of x is 0, so b = 0!

Q: How do I remember the vertex formula?

Think of it as "negative b over 2a" - it finds the x-value exactly halfway between the parabola's roots. When b = 0, the vertex sits right on the y-axis at x = 0.

Q: What if I get confused about which coordinate is which?

Remember: (x,y) means "go right x units, then up y units". For (0,3), you don't move horizontally (x=0) but go up 3 units (y=3). Think "x comes first, y comes second"!

Q: Does every parabola have exactly one symmetry point?

Yes! Every quadratic function has exactly one vertex, which is its symmetry point. The parabola is perfectly symmetric - if you fold it along the vertical line through the vertex, both sides match perfectly.

Quadratic Functions with Vertex Form

Given the expression of the quadratic function

Finding the symmetry point of the function

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the point of symmetry in the function

00:03 Point of symmetry is the point where if you fold the parabola in half

00:06 The halves will be equal to each other

00:11 Let's examine the function's coefficients

00:22 We'll use the formula to calculate the vertex point

00:25 We'll substitute appropriate values according to the given data and solve for X at the point

00:30 This is the X value at the point of symmetry

00:33 Now let's substitute this X value in the function to find the Y value at the point

00:42 This is the Y value at the point of symmetry

00:48 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the expression of the quadratic function

Finding the symmetry point of the function

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

Step-by-step solution

To solve this problem, we need to find the symmetry point of the quadratic function given by $f(x) = 3 + 3x^2$ .

Step 1: Identify the type of quadratic equation. Here, it's $ax^2 + bx + c$ where $a = 3$ , $b = 0$ , and $c = 3$ .
Step 2: Use the formula for the symmetry point $x = -\frac{b}{2a}$ . Since $b = 0$ , the formula simplifies to $x = 0$ .
Step 3: Calculate the y-coordinate by substituting $x = 0$ into the original function: $f(0) = 3(0)^2 + 3 = 3$ .

Thus, the symmetry point (also the vertex of the parabola) is $(0, 3)$ .

Therefore, the solution to the problem is $(0, 3)$ .

Final Answer

$(0,3)$

Key Points to Remember

Essential concepts to master this topic

Symmetry Point: The vertex of a parabola is its symmetry point
Formula: Use $x = -\frac{b}{2a}$ to find x-coordinate of vertex
Check: Substitute x-value back: f(0) = 3 + 3(0)² = 3 gives point (0,3) ✓

Common Mistakes

Avoid these frequent errors

Confusing coordinates and writing (3,0) instead of (0,3)
Don't swap the x and y coordinates = wrong point location! Students often mix up which value goes first in the ordered pair. Always remember (x,y) format: x-coordinate first, then y-coordinate.

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

What exactly is a symmetry point in a quadratic function?

The symmetry point is the vertex of the parabola - the point where the graph is perfectly balanced! For $f(x) = 3 + 3x^2$ , it's the lowest point at (0,3) where the parabola turns around.

Why is b = 0 in this function?

In the standard form $ax^2 + bx + c$ , our function $f(x) = 3 + 3x^2$ has no x term. That means the coefficient of x is 0, so b = 0!

How do I remember the vertex formula?

Think of it as "negative b over 2a" - it finds the x-value exactly halfway between the parabola's roots. When b = 0, the vertex sits right on the y-axis at x = 0.

What if I get confused about which coordinate is which?

Remember: (x,y) means "go right x units, then up y units". For (0,3), you don't move horizontally (x=0) but go up 3 units (y=3). Think "x comes first, y comes second"!

Does every parabola have exactly one symmetry point?

Yes! Every quadratic function has exactly one vertex, which is its symmetry point. The parabola is perfectly symmetric - if you fold it along the vertical line through the vertex, both sides match perfectly.

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Locate the Symmetry Point of the Quadratic Function f(x)=3+3x^2

Quadratic Functions with Vertex Form

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What exactly is a symmetry point in a quadratic function?

Why is b = 0 in this function?

How do I remember the vertex formula?

What if I get confused about which coordinate is which?

Does every parabola have exactly one symmetry point?

🌟 Unlock Your Math Potential

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