Problem 70 - Precalculus : Building Concepts and Connections

Question

Examine the following table of values for a quadratic function \(f\) $$\begin{array}{rr} x & f(x) \\ -2 & 3 \\ -1 & 0 \\ 0 & -1 \\ 1 & 0 \\ 2 & 3 \end{array}$$ (a) What is the equation of the axis of symmetry of the associated parabola? Justify your answer. (b) Find the minimum or maximum value of the function and the value of \(x\) at which it occurs. (c) Sketch a graph of the function from the values given in the table, and find an expression for the function.

Step-by-Step Solution

Answer

The axis of symmetry is \(x=0\), the minimum value is -1, and it occurs at \(x=0\). The quadratic function is \( f(x) = 3/4 * x^2 - 1 \)

Step 1: Find the Axis of Symmetry

The axis of symmetry of a parabola can be found from its turning point (also known as the vertex). The axis of symmetry is the x-coordinate of the turning point. For instance, if (h, k) is the turning point, the equation of the axis is \(x = h\). By examining the given table of values, it is evident that the turning point is (0, -1), because -1 is the smallest f(x) value. Hence, the equation of the axis of symmetry is \(x = 0\).

Step 2: Find the Minimum or Maximum Value

The minimum or maximum value of a quadratic function is the y-coordinate of the turning point. In this case, the turning point is (0, -1), so the minimum value is -1. This value happens at \(x = 0\).

Step 3: Sketch a Graph and Find the Function Expression

Plot all the given points, then sketch a parabola that passes through these points. Considering symmetry and the turning point, the parabola is a minimum turning point parabola, thus it opens upwards and its standard form will be \(f(x) = a(x-h)^2 + k\). Since we already have the turning point (0, -1), thus h = 0 and k = -1. Also, since the graph passes through (-2, 3), substituting these values, we get 3 = 4a which gives \(a = 3/4\). Applying these, the quadratic function is \(f(x) = 3/4 * x^2 - 1\).

Key Concepts

Axis of Symmetry

Understanding the axis of symmetry of a parabola is crucial when analyzing quadratic functions. The axis of symmetry is a vertical line that divides the parabola into two mirror images. For any quadratic function in the form of \(f(x) = ax^2 + bx + c\), the axis of symmetry can be calculated using the formula \(x = -\frac{b}{2a}\).

This line represents a fundamental characteristic of the parabola, passing through its highest or lowest point, known as the vertex. The axis of symmetry provides valuable insights into the graph's structure, allowing us to predict the parabola's behavior without constructing the complete graph.

Vertex of a Parabola

The vertex of a parabola is a key concept in quadratic analysis—it's the peak or the lowest point on the graph, depending on whether the parabola opens upwards or downwards. In mathematical terms, if a parabola is represented by the quadratic function \(f(x) = ax^2 + bx + c\), the coordinates of the vertex can be found using \(x = -\frac{b}{2a}\) and \(f(x)\) for that x-value.

Identifying the vertex helps to determine the maximum or minimum value of the function, and it serves as a pivotal point for symmetry. When plotting a quadratic function, starting by marking the vertex on the graph makes it easier to sketch the rest of the parabola.

Minimum Value of a Quadratic Function

In quadratic functions of the form \(f(x) = ax^2 + bx + c\), when 'a' is positive, the graph opens upwards, indicating the presence of a minimum value. This minimum value is the y-coordinate of the vertex. To find this value, once we've calculated \(x = -\frac{b}{2a}\), we plug this x-value back into the original function. The output \(f(x)\) is the minimum value of the function.

Students often mix up the x-coordinate of the vertex with the minimum value. Remember, the x-coordinate gives us the axis of symmetry, while the y-coordinate provides the actual minimum value of the quadratic function.

Sketching Parabolas

Sketching parabolas is an illustrative way to understand quadratic functions. To sketch a parabola, begin by finding the vertex and plotting it on a coordinate plane. Use the axis of symmetry to help guide the general shape of the parabola.

After plotting the vertex, you can plot additional points, typically by choosing x-values around the vertex. These points should reflect the symmetry across the axis. Drawing a smooth curve through these points will give you the parabola. It is important to note the direction in which the parabola opens. A positive 'a' in the function \(f(x) = ax^2 + bx + c\) means the parabola opens upwards, while a negative 'a' means it opens downwards. Always double-check your sketch for symmetry and correct opening direction.