Problem 20 - Algebra 2

Question

Graph each function. Label the vertex and the axis of symmetry. $$ y=\frac{1}{2} x^{2}+2 x-8 $$

Step-by-Step Solution

Answer

The vertex of the parabola is \((-2, -10)\) and the axis of symmetry is the line \(x = -2\). The parabola opens upwards because \(a=\frac{1}{2}\) is positive.

Step 1: Convert to Vertex Form

Convert the quadratic function to vertex form, which is given by \(y=a(x-h)^{2}+k\), where \((h,k)\) is the vertex of the parabola. Start by factoring the coefficient \(a\) out of the \(x^2\) and \(x\) terms: \(y = \frac{1}{2}(x^2 + 4x) - 8\). Complete the square by adding and subtracting \((\frac{4}{2})^2 = 4\): \(y = \frac{1}{2}(x^2 + 4x + 4 - 4) - 8\), which simplifies to \(y = \frac{1}{2}(x+2)^2 - 10\).

Step 2: Identify the Vertex

In the vertex form \(y=a(x-h)^2+k\), the vertex \((h,k)\) can be directly read from the equation. The vertex of the function \(y = \frac{1}{2}(x+2)^2 - 10\) is \((-2, -10)\).

Step 3: Draw the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. The equation of the axis of symmetry is \(x=h\), where \(h\) is the x-coordinate of the vertex. Therefore, the axis of symmetry for this function is the line \(x = -2\).

Step 4: Sketch the Graph

Graph the parabola by plotting the vertex and several points on either side of the axis of symmetry. Since \(a=\frac{1}{2}\) is positive, the parabola opens upwards. Label the vertex and the axis of symmetry on the graph.

Key Concepts

Vertex Form of a Quadratic

Understanding the vertex form of a quadratic function is essential to graphing parabolas efficiently. The general expression for vertex form is given by \(y = a(x-h)^2 + k\), where \(h, k)\