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. \(f(x)=(x-3)^{2}-2\)

Short answer

Vertex: (3, -2), Axis of Symmetry: x = 3, Concave Up.

Step-by-step solution

Identify the Form of the Quadratic Function

Notice that the quadratic function is given in vertex form, which is \[ f(x) = a(x-h)^2 + k \]. Here, we can identify that \( a = 1 \), \( h = 3 \), and \( k = -2 \).

Find the Vertex

Using the identified values, the vertex of the function is at the point \( (h, k) = (3, -2) \).

Determine the Axis of Symmetry

The axis of symmetry for a quadratic function in vertex form is given by the line \( x = h \). Therefore, the axis of symmetry is \( x = 3 \).

Determine Concavity

The sign of \( a \) determines if the graph is concave up or concave down. If \( a > 0 \), the graph is concave up. If \( a < 0 \), the graph is concave down. Here, \( a = 1 \), which is positive, so the graph is concave up.

Graph the Quadratic Function

Plot the vertex at \( (3, -2) \) and draw the axis of symmetry at \( x = 3 \). Since the graph is concave up, sketch the parabola opening upwards.

Key concepts

concavity

Concavity refers to the direction in which the parabola opens. The sign of the leading coefficient \a\ in the vertex form \ f(x) = a(x - h)^2 + k\ determines the concavity: \ a > 0 \ means the parabola opens upwards (concave up), and \ a < 0 \ means it opens downwards (concave down).

In the given problem, \ a = 1 \, which is greater than 0. Therefore, the graph of the quadratic function \f(x) = (x - 3)^2 - 2\ is concave up. Visualize this as the parabola looking like a smile. Using this information, we can confidently sketch the graph by ensuring the curve opens upwards starting from the vertex \(3, -2\).

Understanding concavity is crucial for predicting the general shape and direction of the parabola, aiding in accurate graph plotting.