Problem 34 - Algebra 2 and Trigonometry

Question

Determine the coordinates of the vertex and the equation of the axis of symmetry of \(\mathrm{f}(x)=x^{2}+8 x+5\) by writing the equation in the form \(\mathrm{f}(x)=(x-h)^{2}+k .\) Justify your answer.

Step-by-Step Solution

Answer

The vertex is (-4, -11), and the axis of symmetry is x = -4.

Step 1: Identify the Standard and Vertex Form

The standard form of a quadratic function is \( ax^2 + bx + c \). For the given function, \( f(x) = x^2 + 8x + 5 \), we want to convert it into the vertex form, \( f(x) = (x-h)^2 + k \), where \((h, k)\) is the vertex.

Step 2: Completing the Square

To convert to the vertex form, first identify the coefficient of \( x \), which is \( 8 \). Take half of this coefficient to complete the square: \( \frac{8}{2} = 4 \). Square this value: \( 4^2 = 16 \). Now rewrite the function, adding and subtracting 16 inside the function: \( f(x) = x^2 + 8x + 16 - 16 + 5 \).

Step 3: Rewriting in Vertex Form

The expression \( x^2 + 8x + 16 \) is a perfect square trinomial, which can be written as \((x+4)^2\). Thus, the function becomes: \( f(x) = (x+4)^2 - 11 \). This is now in vertex form, where \( h = -4 \) and \( k = -11 \).

Step 4: Determine Vertex and Axis of Symmetry

From the vertex form \( f(x) = (x-h)^2 + k \), the vertex \((h, k)\) is found. Here, it is \((-4, -11)\). The axis of symmetry is the vertical line through the vertex, given by \( x = -4 \).

Key Concepts

Completing the Square

Completing the square is a technique used to express a quadratic equation in vertex form. The goal is to transform an equation from its standard form, \(ax^2 + bx + c\), into the form \((x-h)^2 + k\). This makes it easier to identify key features like the vertex of the parabola. Here's how to do it:

• Identify the coefficient of \(x\), which is the \(b\) term in the standard form.
• Take half of this coefficient and square it. This value is added and subtracted inside the equation to form a perfect square trinomial.

For example, in the function \(f(x) = x^2 + 8x + 5\), the coefficient of \(x\) is \(8\). Half of \(8\) is \(4\), and \(4^2 = 16\). So, you add and subtract \(16\) to the function: \(f(x) = x^2 + 8x + 16 - 16 + 5\). Now, the \(x^2 + 8x + 16\) part can be rewritten as \((x + 4)^2\), allowing you to complete the square.

Axis of Symmetry

The axis of symmetry is a line that divides a parabola into two mirror-image halves. This is a critical feature of quadratic functions as it helps in analyzing the function's behavior and graph. Once you've written a quadratic function in vertex form, \((x-h)^2 + k\), the axis of symmetry can be easily identified.

• The axis of symmetry is given by the equation \(x = h\), where \(h\) is the x-coordinate of the vertex.
• For our function, once transformed to vertex form, we found it to be \((x + 4)^2 - 11\). This means that \(h = -4\), so the axis of symmetry is \(x = -4\).

This vertical line at \(x = -4\) splits the parabola down the middle, and any point on one side has a mirror point on the other side.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically written in the standard form \(ax^2 + bx + c\). These functions graph to form parabolas, which can open upwards or downwards depending on the sign of the leading coefficient, \(a\).

• If \(a > 0\), the parabola opens upwards, and its vertex is the minimum point.
• If \(a < 0\), it opens downwards, and the vertex is the maximum point.

The vertex form \((x-h)^2 + k\) is especially useful because it directly provides the parabola's vertex, \((h, k)\). Knowing the vertex helps in graphing the quadratic function quickly, finding its maximum or minimum, and understanding its symmetry. In our example, after converting to vertex form, we determined the vertex to be \((-4, -11)\), indicating the lowest point if the parabola opens upwards as \(a > 0\).