Find the Axis of Symmetry f(x)=x^2+8x
The given problem asks to determine the axis of symmetry for the quadratic function f(x) = x^2 + 8x. The axis of symmetry of a quadratic function is a vertical line that bisects the parabola into two mirror-image halves. For any quadratic function in standard form (f(x) = ax^2 + bx + c), the axis of symmetry can be found using the formula x = -b/(2a). The task is to apply this formula to the provided quadratic equation to find the specific value of x that represents the axis of symmetry for the graph of the given function.
Express the function
Transform the quadratic equation into the vertex form.
Complete the square for the expression
Identify the coefficients
Recall the vertex form of a quadratic equation:
Calculate the value of
Insert the known values for
Simplify the fraction by reducing common factors.
Extract the factor of 2 from the numerator.
Eliminate the common factors.
Factor out 2 from the denominator.
Remove the common factor of 2.
Rewrite the simplified expression.
Perform the division of 4 by 1.
Determine the value of
Place the values of
Simplify the expression on the right side.
Simplify each component.
Square the number 8.
Multiply 4 by 1.
Divide 64 by 4.
Multiply -1 by 16.
Subtract 16 from 0.
Insert the calculated values of
Set
Utilize the vertex form
As the coefficient
Identify the vertex
Compute
Use the formula
Substitute the value of
Eliminate the common factor of 1.
Determine the focus of the parabola.
To find the focus, add
Insert the known values for
The axis of symmetry is the line that passes through both the vertex and the focus.
The axis of symmetry for the given quadratic function
To find the axis of symmetry for a quadratic function, one can use the vertex form of a quadratic equation, which is
The process of completing the square involves manipulating a quadratic equation into the vertex form. This is done by adding and subtracting a particular value inside the squared term to create a perfect square trinomial, which then simplifies to the vertex form.
The vertex form provides valuable information about the parabola, including the direction it opens (upward for
The focus of a parabola is a fixed point used in the geometric definition of the curve and is related to the vertex by the distance
In this problem, we used algebraic manipulation to rewrite the given quadratic function in vertex form, from which we could easily determine the axis of symmetry.