Problem 50 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\) in the viewing rectangle $$ [-4,7,4,7,1] \text { by }[-3,1,3,1,1] $$ $$ f(x)=3-1.5 x^{2} $$

Step-by-Step Solution

Answer

Graph the parabola within the viewing rectangle by plotting key points and noting it extends beyond limits.

Step 1: Set the Domain and Range

Identify the domain and range based on the viewing rectangle. The domain is given by the x-coordinates i.e., [-4, 7] with a scale marking every 1 unit. The range for the y-values is [-3, 1] with a scale marking every 1 unit.

Step 2: Calculate Key Points

Evaluate the function at several key points within the domain. For example:- When x = -4, since: \[f(-4) = 3 - 1.5(-4)^2 = 3 - 24 = -21\]- When x = 0, since: \[f(0) = 3 - 1.5(0)^2 = 3\]- When x = 7, since: \[f(7) = 3 - 1.5(7)^2 = 3 - 73.5 = -70.5\]This shows that the function takes on values outside the plotting range specified by the viewing rectangle.

Step 3: Plot Basic Curve

Using key points select values within the graph's range to plot a few visible points inside the viewing rectangle. Evaluate at additional convenient points like x = -2, x = 2, and x = 3:- When x = -2, since: \[f(-2) = 3 - 1.5(-2)^2 = 3 - 6 = -3\]- When x = 2, since: \[f(2) = 3 - 1.5(2)^2 = 3 - 6 = -3\]- When x = 3, since: \[f(3) = 3 - 1.5(3)^2 = 3 - 13.5 = -10.5\]Plot these points and draw the parabolic curve.

Step 4: Draw the Graph

Plot all calculated points, connect them to form a parabola. Note the shape is a downward-opening parabola based on the coefficient -1.5 in front of \(x^2\).

Step 5: Analyze and Adjust Graph

Ensure that the plotted graph lies within the specified viewing rectangle and correct for visual discrepancies. Note that the maximum point at \(x = 0\) with \(f(0) = 3\) and other calculated points allows for an approximate visualization of how the parabola extends beyond the visible range.

Key Concepts

Domain and Range

The domain of a function refers to all the possible input values (often x-values) that you can use to evaluate the function without running into any issues like division by zero or taking the square root of a negative number. Meanwhile, the range consists of all the possible output values (or y-values) that result from using these inputs.
The exercise gives us a viewing rectangle to work within, specifically the domain

• Domain: \([-4, 7]\)
• Range: \([-3, 1]\)

This tells us that we're concentrating on x-values from -4 to 7 and y-values from -3 to 1. Even though the quadratic function can take on values outside of this range, when plotting our graph, we need to focus only on this section to abide by the limits given. It helps us to plot effectively in practical scenarios where we want to see specific parts of the graph.

Quadratic Functions

A quadratic function is a type of polynomial function characterized by the equation: \[ f(x) = ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants, and \(a\) is non-zero. In this exercise, the quadratic function is given as \[ f(x) = 3 - 1.5x^2 \]This function has no \(bx\) term, simplifying our graphing task. The graph of any quadratic function is a U-shaped curve called a parabola. Here, because \(a = -1.5\) is negative, the parabola opens downward, indicating that as you move away from its vertex, the graph will fall off, moving towards negative infinity
The vertex represents either the maximum or minimum point. In this case, it is a maximum because of the downward orientation. The vertex of \[ f(x) = 3 - 1.5x^2 \]is at \((0, 3)\), being the highest point on this viewing rectangle. This information helps in sketching the initial shape of the parabola, assisting in determining how it behaves, and finding where key points might lie.

Plotting Key Points

Once you understand the domain, range, and general shape of the graph, the next step is to determine key points that lie along the curve. Key points provide reference points making the drawing of the curve simpler and more accurate. To find key points:

• Start from the middle. For this function, \(x = 0\) gives us \(f(0) = 3\), the vertex and highest point.
• Check both ends. With \(x = -4\) and \(x = 7\), you discover that these coordinates, while important, lead to values outside the viewing range.
• Calculate additional points. Values within the range, like \(x = -2\), \(2\), and \(3\), are significant since \(f(-2)\) and \(f(2) = -3\) sit right at the range's edge.

Plot these points on your graph, and once these are in place, you can draw a smooth curve through these plotted points to form the parabola. It is essential to reflect changes in direction that occur at the vertex.