Determine the Axis of Symmetry for 3x^2 + 6x - 6

Quadratic Functions with Axis of Symmetry

Calculate the axis of symmetry of the quadratic function below:

f(x)=3x2+6x−6 f(x)=3x^2+6x-6

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the axis of symmetry for the function
00:03 The axis of symmetry is the X value at the vertex point
00:06 The point where if you fold the parabola in half, both halves are identical
00:09 Let's look at the function's coefficients
00:18 We'll use the formula to calculate the vertex point
00:24 We'll substitute appropriate values according to the given data and solve for X point
00:38 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

Calculate the axis of symmetry of the quadratic function below:

f(x)=3x2+6x−6 f(x)=3x^2+6x-6

2

Step-by-step solution

To find the axis of symmetry for the quadratic function f(x)=3x2+6x−6 f(x) = 3x^2 + 6x - 6 , we begin by identifying the coefficients in the general form of a quadratic equation: ax2+bx+c ax^2 + bx + c . Here, a=3 a = 3 , b=6 b = 6 , and c=−6 c = -6 .

The formula for the axis of symmetry of a quadratic function is:

x=−b2a x = -\frac{b}{2a} .

Substituting the given values into the formula, we have:

x=−62⋅3 x = -\frac{6}{2 \cdot 3} .

Calculating the above expression, we get:

x=−66=−1 x = -\frac{6}{6} = -1 .

Thus, the axis of symmetry for this quadratic function is x=−1 x = -1 .

Therefore, the solution to the problem is x=−1 x = -1 .

3

Final Answer

x=−1 x=-1

Key Points to Remember

Essential concepts to master this topic
  • Formula: Use x=−b2a x = -\frac{b}{2a} for any quadratic function
  • Technique: Identify coefficients: a = 3, b = 6, then calculate −62(3)=−1 -\frac{6}{2(3)} = -1
  • Check: Vertex x-coordinate should equal axis value: both are -1 ✓

Common Mistakes

Avoid these frequent errors
  • Using wrong signs in the axis formula
    Don't forget the negative sign in x=−b2a x = -\frac{b}{2a} = wrong axis location! Students often write x=b2a x = \frac{b}{2a} and get positive 1 instead of negative 1. Always include the negative sign before the fraction.

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 exactly is the axis of symmetry?

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The axis of symmetry is a vertical line that divides the parabola into two mirror halves. For f(x)=3x2+6x−6 f(x) = 3x^2 + 6x - 6 , the line x=−1 x = -1 splits the curve perfectly down the middle.

Why do we use the formula x=−b2a x = -\frac{b}{2a} ?

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This formula comes from completing the square or using calculus to find where the derivative equals zero. It always gives you the x-coordinate of the vertex, which is exactly where the axis of symmetry passes through.

What if my quadratic doesn't have a middle term?

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If there's no x term (like y=2x2+5 y = 2x^2 + 5 ), then b = 0. Using the formula: x=−02a=0 x = -\frac{0}{2a} = 0 , so the axis of symmetry is the y-axis!

Can I find the axis without memorizing the formula?

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Yes! You can complete the square to rewrite the function in vertex form f(x)=a(x−h)2+k f(x) = a(x - h)^2 + k . The axis of symmetry is always x=h x = h .

How do I know if my axis is correct?

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Check that points on opposite sides of your axis have the same y-value! For x=−1 x = -1 , try x=0 x = 0 and x=−2 x = -2 : both should give f(x)=−6 f(x) = -6 .

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