Problem 38 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of symmetry. $$ F(x)=\left(x+\frac{1}{2}\right)^{2}-2 $$

Step-by-Step Solution

Answer

Vertex: \((-\frac{1}{2}, -2)\); Axis: \(x = -\frac{1}{2}\); Upward-opening parabola.

Step 1: Identify the Standard Form

The given quadratic function is \( F(x) = \left(x+\frac{1}{2}\right)^2-2 \). This is already in vertex form \( y = a(x-h)^2 + k \). By comparing, we see that \( h = -\frac{1}{2} \) and \( k = -2 \).

Step 2: Determine the Vertex

The vertex of a quadratic in the form \( y = a(x-h)^2 + k \) is \( (h, k) \). From Step 1, we have \( h = -\frac{1}{2} \) and \( k = -2 \), so the vertex is \( \left(-\frac{1}{2}, -2\right) \).

Step 3: Identify the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line passing through the vertex. Since the x-coordinate of the vertex is \( -\frac{1}{2} \), the axis of symmetry is the line \( x = -\frac{1}{2} \).

Step 4: Sketch the Graph

Plot the vertex \((-\frac{1}{2}, -2)\) on a graph. Draw a vertical dashed line at \(x = -\frac{1}{2}\) for the axis of symmetry. Since the coefficient of \((x + \frac{1}{2})^2\) is positive, the parabola opens upwards. Sketch the curve symmetrically around the axis of symmetry. Label the vertex and the axis.

Key Concepts

Vertex Form

The vertex form of a quadratic function is an important way to express a quadratic equation, especially when you want to easily identify the vertex of the parabola it represents. It has the general structure: \[ y = a(x-h)^2 + k \] where:

• \(a\) determines the "width" and the "direction" of the parabola (upward if \(a > 0\), downward if \(a < 0\)).
• \(h\) and \(k\) represent the x and y coordinates of the parabola's vertex, respectively.

This form is particularly beneficial because it gives a clear and immediate view of the vertex. Given the function \( F(x) = \left(x + \frac{1}{2}\right)^2 - 2 \), which is already in vertex form, you can quickly identify its vertex at \( \left(-\frac{1}{2}, -2\right) \).
Moving from standard form or factored form to vertex form often involves a process called completing the square. The vertex form is a friendly tool in graphing parabolas because it allows you to graph a key feature of the parabola, the vertex, without needing to find it separately.

Axis of Symmetry

Understanding the axis of symmetry is crucial when learning about parabolas. The axis of symmetry is a vertical line that splits the parabola into two mirror-image halves. For a quadratic function in vertex form \( y = a(x-h)^2 + k \), the axis of symmetry is given by the equation \( x = h \).
In the given exercise, where the quadratic function is \( F(x) = \left(x + \frac{1}{2}\right)^2 - 2 \), the vertex is \( \left(-\frac{1}{2}, -2\right) \). Here, the axis of symmetry is the vertical line \( x = -\frac{1}{2} \) because that is the x-coordinate of the vertex.

• It is a simple line which helps us understand how the parabola is symmetric.
• The parabola reflects over this line, meaning for every point \((x, y)\) on the parabola, there is a matching point \((-x, y)\) on the other side.

This axis is a helpful guide when sketching the parabola, especially when visualizing how the graph behaves and ensuring that the two sides of the parabola are symmetric around the line.

Graphing Parabolas

Graphing parabolas involves plotting the key features derived from the quadratic equation. These include the vertex, axis of symmetry, and knowing the direction the parabola opens. Using vertex form to graph a parabola is a straightforward process.
Start by plotting the vertex of the parabola, which gives you a fixed starting point. In our example, the vertex of the function \( F(x) = \left(x + \frac{1}{2}\right)^2 - 2 \) is at \( (-\frac{1}{2}, -2) \). Mark this point on your graph.

• Next, draw the axis of symmetry, a vertical dashed line passing through \( x = -\frac{1}{2} \).
• After pinpointing these elements, check the coefficient \( a \) from the equation to know the direction of the parabola. Since \( a \) is positive, the parabola opens upwards.

Once the vertex and direction are known, graphing involves sketching a curve that mirrors itself across the axis of symmetry. The more points you plot symmetrically across the axis, the more accurate your parabola will be. Remember, the symmetry is key to ensuring that your graph is the true reflection of the quadratic function you are working with.