Determine the Axis of Symmetry of the Quadratic Function f(x) = x^2 + 4x + 1

Quadratic Functions with Axis Formula

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

f(x)=x2+4x+1 f(x)=x^2+4x+1

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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
00:10 Let's examine the function coefficients
00:18 We'll use the formula to calculate the vertex point
00:23 We'll substitute appropriate values according to the given data and solve for X at the point
00:34 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Given the expression of the quadratic function

The symmetrical axis of the expression must be found.

f(x)=x2+4x+1 f(x)=x^2+4x+1

2

Step-by-step solution

To solve this problem, we'll apply the formula for finding the axis of symmetry for a quadratic function:

  • Step 1: Identify the coefficients.
    For the quadratic function f(x)=x2+4x+1 f(x) = x^2 + 4x + 1 , we have a=1 a = 1 and b=4 b = 4 .
  • Step 2: Use the formula for the axis of symmetry.
    The axis of symmetry for a quadratic function ax2+bx+c ax^2 + bx + c is given by x=−b2a x = -\frac{b}{2a} .
  • Step 3: Substitute the values of a a and b b into the formula.
    Substitute b=4 b = 4 and a=1 a = 1 to get: x=−42×1 x = -\frac{4}{2 \times 1} .
  • Step 4: Simplify to find the value of x x .
    This simplifies to x=−42=−2 x = -\frac{4}{2} = -2 .

Therefore, the axis of symmetry for the given quadratic function is x=−2 x = -2 .

3

Final Answer

x=−2 x=-2

Key Points to Remember

Essential concepts to master this topic
  • Formula: Axis of symmetry is x=−b2a x = -\frac{b}{2a} for standard form
  • Technique: From x2+4x+1 x^2 + 4x + 1 , use a=1, b=4: x=−42(1)=−2 x = -\frac{4}{2(1)} = -2
  • Check: Vertex x-coordinate should equal axis value: f(-2) gives vertex ✓

Common Mistakes

Avoid these frequent errors
  • Using wrong sign in the axis formula
    Don't forget the negative sign in x=−b2a x = -\frac{b}{2a} = wrong axis location! Students often write x=42=2 x = \frac{4}{2} = 2 instead of x=−42=−2 x = -\frac{4}{2} = -2 . 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

Why is the axis of symmetry formula negative?

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The formula x=−b2a x = -\frac{b}{2a} comes from completing the square! The negative sign ensures the parabola's vertex is positioned correctly relative to the linear term coefficient.

What if the coefficient 'a' is negative?

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The formula still works! If a = -2 and b = 4, then x=−42(−2)=−4−4=1 x = -\frac{4}{2(-2)} = -\frac{4}{-4} = 1 . Just be extra careful with the signs when calculating.

How can I verify my axis of symmetry is correct?

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Check that points equidistant from the axis have the same y-value. For axis x = -2, try x = -3 and x = -1: both should give f(x) = -2!

Does the axis of symmetry depend on the constant term 'c'?

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No! The constant term doesn't affect the axis position - only coefficients 'a' and 'b' matter. The axis formula x=−b2a x = -\frac{b}{2a} doesn't include 'c'.

Can the axis of symmetry be a fraction?

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Absolutely! If you have f(x)=x2+3x+1 f(x) = x^2 + 3x + 1 , the axis would be x=−32=−1.5 x = -\frac{3}{2} = -1.5 . Fractional axes are completely normal.

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