Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=x^{2}+8\)

Step-by-Step Solution

Answer

The graph of function \(f(x)=x^{2}+8\) is a parabola that opens upward. The vertex of the parabola is at the point (0,8).

Step 1: Identify the vertex

The vertex form of a quadratic function is \(f(x)=a(x-h)^{2}+k\). Comparing this with the given function \(f(x)=x^2+8\), it can be observed that \(h = 0\) and \(k = 8\). Since the coefficient of \(x^2\) is positive, the parabola opens upwards. Meanwhile, the value of \(k\) indicates a vertical shift upwards by 8 units. Therefore, the vertex of the function is at the point (0,8).

Step 2: Sketch the graph

The graph of the function \(f(x)=x^{2}+8\) is a parabola that opens upward and the vertex is at the point (0,8). The graph has its y-intercept and axis of symmetry at x=0.

Step 3: Use a graphing utility

Utilize a graphing utility software or website to graph the function \(f(x)=x^{2}+8\) and verify the position of the vertex and the shape of the graph. The graph should coincide with the previously identifed features.

Key Concepts

Vertex Identification

In a quadratic function, the vertex is a key point that characterizes the graph's shape and direction. It's the highest or lowest point on the graph of a parabola, depending on whether it opens upwards or downwards. To find the vertex, you can use the vertex form of a quadratic equation: \[ f(x) = a(x-h)^2 + k \] Here,

• \( h \) is the x-coordinate of the vertex
• \( k \) is the y-coordinate of the vertex

For the quadratic function \( f(x) = x^2 + 8 \), we can see that

• \( a = 1 \) (indicating the parabola opens upwards)
• \( h = 0 \)
• \( k = 8 \)

Thus, the vertex is located at the point \( (0, 8) \). The vertex not only indicates a vertical shift in the graph but also serves as a significant reference for the symmetry and direction of the parabola.

Parabola

A parabola is a U-shaped curve that represents the graph of a quadratic function, such as \( f(x) = x^2 + 8 \). Parabolas display a characteristic symmetry around a certain line, which we'll discuss soon. They open either upwards or downwards, depending on the sign of the coefficient \( a \) in the quadratic function.

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

For our function \( f(x) = x^2 + 8 \), \( a = 1 \), which is positive, indicating that the parabola will open upwards. This feature means that the vertex \( (0,8) \) is the lowest point on this graph. The upward-opening parabola implies that as the x-values move away from the vertex in either direction, the y-values increase, creating the distinctive U-shape. The spread or "width" of the parabola depends on the absolute value of \( a \); the larger \( a \), the narrower the parabola.

Axis of Symmetry

The axis of symmetry is an essential aspect of a parabola. It is a vertical line that divides the parabola into two mirror-image halves. For any quadratic function in the standard form \( f(x) = ax^2 + bx + c \), the axis of symmetry can be found using the formula: \[ x = -\frac{b}{2a} \] In the function \( f(x) = x^2 + 8 \), there is no linear x-term (\(b = 0\)), simplifying our calculation to:\[ x = -\frac{0}{2 \times 1} = 0 \] Thus, the axis of symmetry is the vertical line \( x = 0 \), which neatly divides the parabola into two equal parts, with each side mirroring the other. The significance of this symmetry is that it confirms the positions of the vertex and helps verify the direction of the parabola. Knowing the axis of symmetry allows us to understand the even distribution of points around it and effectively predict the graph's behavior.