Question

(a) find the vertex and axis of symmetry of each quadratic function. (b) Determine whether the graph is concave up or concave down. (c) Graph the quadratic function. \(f(x)=-\frac{1}{3}\left(x-\frac{1}{2}\right)^{2}-\frac{7}{6}\)

Short answer

The vertex is \( \left( \frac{1}{2}, -\frac{7}{6} \right) \), the axis of symmetry is \( x = \frac{1}{2} \), and the graph is concave down.

Step-by-step solution

Identify Vertex Form of the Quadratic Function

The given quadratic function is already in vertex form: \[ f(x) = -\frac{1}{3}\left(x - \frac{1}{2}\right)^2 - \frac{7}{6} \].In this form, \( f(x) = a(x - h)^2 + k \), the vertex \( h, k \) is easily identified.

Determine the Vertex

In the equation \( f(x) = -\frac{1}{3}\left(x - \frac{1}{2}\right)^2 - \frac{7}{6} \), the vertex \( h, k \) is \( \left( \frac{1}{2}, -\frac{7}{6} \right) \). This comes from the values of 'h' and 'k' in the vertex form.

Identify the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. Since the vertex is \( \left( \frac{1}{2}, -\frac{7}{6} \right) \), the axis of symmetry is \( x = \frac{1}{2} \).

Determine Concavity

The coefficient 'a' in the standard form \( f(x) = a(x - h)^2 + k \) determines the concavity of the graph. If 'a' is positive, the graph is concave up; if 'a' is negative, it is concave down. Here, \( a = -\frac{1}{3} \), which is negative, so the graph is concave down.

Graph the Quadratic Function

To graph the quadratic function, plot the vertex \( \left( \frac{1}{2}, -\frac{7}{6} \right) \). Draw the axis of symmetry at \( x = \frac{1}{2} \) and sketch the graph opening downwards, ensuring it reflects the concavity derived in the previous step.

Key concepts

Vertex Form

The quadratic function given in the problem is already in vertex form, which is written as: \[f(x) = a(x - h)^2 + k\].

This form is very useful because it directly reveals the vertex of the quadratic function.

In this format, the variables \(h\) and \(k\) represent the coordinates of the vertex. The vertex \(h, k\) is a critical point on the graph where the function reaches its highest or lowest value, depending on whether the parabola opens upwards or downwards.

In this exercise, the function is: \[f(x) = -\frac{1}{3}(x - \frac{1}{2})^2 - \frac{7}{6}\].

So, \(h = \frac{1}{2}\) and \(k = -\frac{7}{6}\). Therefore, the vertex of the function is \((\frac{1}{2}, -\frac{7}{6})\). This point is significant because it helps us graph the function accurately. Additionally, this vertex form makes it simple to identify transformations like shifts and stretches applied to the basic \(x^2\) graph.

Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that divides the parabola into two mirror-image halves.

This line always passes through the vertex. Hence, it directly depends on the vertex's \(h\)-coordinate.

If the vertex of our function is \((\frac{1}{2}, -\frac{7}{6})\), then the axis of symmetry is at \(x = \frac{1}{2}\). This vertical line serves as a reference to ensure our graph is symmetrical.

For any quadratic function in vertex form \(f(x) = a(x - h)^2 + k\), the axis of symmetry will always be \(x = h\). Understanding this helps a great deal in sketching and analyzing the function because it aligns everything around this central dividing line.

Concavity

Concavity describes the direction in which a parabola opens. It is determined by the coefficient \(a\) in the vertex form of the quadratic function \(f(x) = a(x - h)^2 + k\).

If \(a\) is positive, the parabola opens upwards, and we say the graph is concave up. Conversely, if \(a\) is negative, the parabola opens downwards, making the graph concave down.

In our function \[f(x) = -\frac{1}{3}(x - \frac{1}{2})^2 - \frac{7}{6}\], \(a = -\frac{1}{3}\). Since this coefficient is negative, the graph of the function is concave down. This means it resembles an upside-down 'U' shape, with the vertex being the highest point on the graph.

Understanding concavity helps us decide how to sketch the function accurately and predict its behavior, like where it increases or decreases.