Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Match each function in Column I with the description of the parabola that is its graph in Column II. \(\mathbf{I}\) (a) \(f(x)=(x-4)^{2}-2\) (b) \(f(x)=(x-2)^{2}-4\) (c) \(f(x)=-(x-4)^{2}-2\) (d) \(f(x)=-(x-2)^{2}-4\) \(\mathbf{I I}\) A. Vertex \((2,-4),\) opens downward B. Vertex \((2,-4),\) opens upward C. Vertex \((4,-2),\) opens downward D. Vertex \((4,-2),\) opens upward

Step-by-Step Solution

Answer

(a) D, (b) B, (c) C, (d) A.

Step 1: Analyze Function a

The function given is \( f(x) = (x-4)^2 - 2 \). It is in the vertex form \( f(x) = (x-h)^2 + k \), where the vertex is \( (h, k) \). For this function, \( h = 4 \) and \( k = -2 \), so the vertex is \( (4, -2) \). The parabola opens upward because the coefficient of \( (x-4)^2 \) is positive.

Step 2: Analyze Function b

The function given is \( f(x) = (x-2)^2 - 4 \). It is in the vertex form \( f(x) = (x-h)^2 + k \), where the vertex is \( (h, k) \). For this function, \( h = 2 \) and \( k = -4 \), so the vertex is \( (2, -4) \). The parabola opens upward because the coefficient of \( (x-2)^2 \) is positive.

Step 3: Analyze Function c

The function is \( f(x) = -(x-4)^2 - 2 \). It follows the vertex form \( f(x) = -(x-h)^2 + k \), where the vertex is \( (h, k) \). Here, \( h = 4 \) and \( k = -2 \), so the vertex is \( (4, -2) \). The parabola opens downward due to the negative coefficient \( -(x-4)^2 \).

Step 4: Analyze Function d

The function is \( f(x) = -(x-2)^2 - 4 \). It is in the vertex form \( f(x) = -(x-h)^2 + k \) with vertex \( (h, k) \). In this case, \( h = 2 \) and \( k = -4 \), so the vertex is \( (2, -4) \). The parabola opens downward because of the negative coefficient \( -(x-2)^2 \).

Step 5: Match Functions to Descriptions

Based on the analysis: Function (a) matches with D (Vertex (4, -2), opens upward). Function (b) matches with B (Vertex (2, -4), opens upward). Function (c) matches with C (Vertex (4, -2), opens downward). Function (d) matches with A (Vertex (2, -4), opens downward).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a powerful way to express and analyze parabolas. It is written as \( f(x) = a(x-h)^2 + k \). In this form, any quadratic function highlights the vertex of the parabola, which is the point \((h, k)\). This makes identifying the vertex straightforward and enables us to quickly determine the parabola's orientation.Some key points about the vertex form:

• Vertex: The vertex \((h, k)\) is the turning point of the parabola. Depending on whether the parabola opens upwards or downwards, the vertex is either the lowest or highest point.
• Coefficient \(a\): This number tells us the direction and width of the parabola. If \(a > 0\), the parabola opens upward, while if \(a < 0\), it opens downward. A larger absolute value of \(a\) makes the parabola narrower, whereas a smaller one makes it wider.

Using vertex form is essential when graphing parabolas because it provides immediate insights into the structure and position of the quadratic graph.

Graphing Parabolas

When graphing parabolas, the vertex form of the quadratic function helps enormously. Knowing the vertex allows you to place the key point of the parabola on the graph accurately. Here's how you can graph a parabola using the vertex form.

• Identify the Vertex: Extract the \((h, k)\) from the function. This will be your starting point on the graph.
• Determine the Axis of Symmetry: The line \(x = h\) acts as the axis of symmetry. The parabola is mirrored across this vertical line.
• Decide the Direction: Look at the sign of the coefficient \(a\). If it's positive, sketch the arms of the parabola opening upward, forming a "U" shape. If it's negative, they turn downward, forming an inverted "U".
• Plot Additional Points: To make your graph more accurate, select a few x-values around the vertex and calculate the corresponding y-values. Plot these points and connect them smoothly.

By following these steps, you can graph any parabola efficiently and accurately, allowing you to visualize the quadratic function's behavior.

Quadratic Functions

Quadratic functions come in the form \( f(x) = ax^2 + bx + c \), and they express some of the most fundamental relationships in algebra. The graph of a quadratic function is a parabola, a symmetrical curve that can open upwards or downwards.Different forms of quadratic functions are useful in various mathematical contexts:

• Standard Form: \(ax^2 + bx + c\) is useful for identifying the y-intercept, \(c\), directly. It is less intuitive for finding the vertex but very common in mathematical equations and solutions.
• Vertex Form: As mentioned, \(f(x) = a(x-h)^2 + k\) offers immediate information about the vertex and direction of the parabola, making analysis simpler.
• Factored Form: This involves expressing the function as \(a(x-r)(x-s)\), where \(r\) and \(s\) are the roots. This form is optimal for solving equations since it directly provides the function's zeros or x-intercepts.

Quadratic functions are not only crucial in algebra but also in calculus, physics, and engineering, often describing projectile motion and other naturally occurring parabolic relationships.