Problem 48 - Precalculus : Building Concepts and Connections

Question

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$f(x)=10 x^{2}-65$$

Step-by-Step Solution

Answer

The vertex of the parabola is (0, -65) and the axis of symmetry is the line \(x = 0\). Two additional points on the graph are (1, -55) and (-1, -55). The sketch of the parabola opens upwards, centered around \(x = 0\).

Step 1: Determine the Vertex

The standard form of a quadratic equation is \(f(x) = a(x-h)^{2} + k\), where \((h, k)\) is the vertex. Given function \(f(x) = 10 x^{2} - 65\) is in the form of \(f(x) = ax^{2} + k\), where \(h = 0\). Therefore, the vertex of the parabola is at (0, -65).

Step 2: Determine the Axis of Symmetry

The axis of symmetry for a parabola is the vertical line \(x = h\), where \(h\) is the x-coordinate of the vertex. In this exampe, \(h = 0\), hence, the axis of symmetry is the line \(x = 0\).

Step 3: Find Two Additional Points

Two additional points on the parabola can be found by substituting any arbitrary values for \(x\) in the given equation. Let's select \(x = 1\) and \(x = -1\). For \(x = 1\), \(f(1) = 10(1)^{2} - 65 = -55\). For \(x = -1\), \(f(-1) = 10(-1)^{2} - 65 = -55\). Therefore, the two additional points are (1, -55) and (-1, -55).

Step 4: Sketch the Parabola

To sketch the graph, visualize a vertical line \(x = 0\) which is the axis of symmetry. Then plot the vertex at (0, -65) and additional points at (1, -55) and (-1, -55). The resulting graph is a parabola opening upwards centred around \(x = 0\).

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is vital since it represents the highest or lowest point of the curve, depending on whether the parabola opens upwards or downwards. In the equation of a quadratic function expressed as f(x) = a(x-h)^{2} + k, the point (h, k) denotes the vertex. For the given function f(x) = 10x^{2} - 65, it simplifies to the vertex being at (0, -65) since there is no x-term to shift the vertex horizontally, meaning h = 0, and k represents the vertical shift of the vertex -65.

When graphing, start by marking the vertex on the coordinate plane. Recognize that it's either the peak or the trough of the parabola; for our example, as a > 0, the parabola opens upwards making (0, -65) the lowest point.

Axis of Symmetry

A parabola is symmetric about a vertical line called the axis of symmetry, which runs through the vertex. The general form to find the axis of symmetry is x = h, where h is the x-coordinate of the vertex. For the function f(x) = 10x^{2} - 65, the vertex (h, k) is (0, -65). Consequently, the axis of symmetry is x = 0.

This line can be thought of as a mirror, where each point on the parabola has a corresponding point equidistant on the opposite side. When plotting multiple points for accuracy, choose x-values that are equidistant from the axis of symmetry to find symmetric points on the parabola.

Quadratic Function Graphing

Finding Additional Points

To sketch a parabola accurately, besides the vertex, you need at least two more points. Choose x-values and plug them into the given quadratic equation to find corresponding y-values. For f(x) = 10x^{2} - 65, selecting x = 1 and x = -1 gives us (1, -55) and (-1, -55), respectively.

Drawing the Parabola

With points in hand, plot them on a graph, ensuring you draw the axis of symmetry. For positive a, the parabola opens upwards, with the vertex being the lowest point. For negative a, it opens downwards, with the vertex being the highest point. The shape should be symmetric and smooth, resembling a U-shape for our positive a function.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is expressed as f(x) = ax^{2} + bx + c. In this, a, b, and c are coefficients that determine the parabola's shape and position. The term a affects the width and direction of the parabola. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards. The term b impacts the horizontal placement of the vertex without affecting the axis of symmetry, while c represents the y-intercept.

Our example, f(x) = 10x^{2} - 65, already simplifies such that b = 0 and c = -65. This function's standard form tells us the parabola opens upward since a = 10 is positive, widens less due to the large value of a, and intersects the y-axis at (0, -65).