Problem 19 - Algebra and Trigonometry

Question

Draw the graphs of \(f, g,\) and \(f+g\) on a common screen to illustrate graphical addition. $$ f(x)=x^{2}, \quad g(x)=\frac{1}{3} x^{3} $$

Step-by-Step Solution

Answer

Graph \( f(x) \), \( g(x) \), and \( f(x) + g(x) \) together; \( f(x) + g(x) \) combines shapes of both functions.

Step 1: Understand the Functions

Before we draw the graphs, let's understand the functions given. The function \( f(x) = x^2 \) is a quadratic function with a parabolic shape opening upwards. The function \( g(x) = \frac{1}{3}x^3 \) is a cubic function with an S-shaped curve.

Step 2: Calculate Points for Function \( f(x) \)

Select values of \( x \) to compute corresponding \( y \) values for \( f(x) = x^2 \). For example: \(-2: f(-2) = 4\), \(-1: f(-1) = 1\), \(0: f(0) = 0\), \(1: f(1) = 1\), \(2: f(2) = 4\). Plot these points on the graph and connect them to form the parabola.

Step 3: Calculate Points for Function \( g(x) \)

Similarly, select values of \( x \) to compute corresponding \( y \) values for \( g(x) = \frac{1}{3}x^3 \). For example: \(-2: g(-2) = -\frac{8}{3}\), \(-1: g(-1) = -\frac{1}{3}\), \(0: g(0) = 0\), \(1: g(1) = \frac{1}{3}\), \(2: g(2) = \frac{8}{3}\). Plot these points on the same graph and connect them to form the cubic curve.

Step 4: Calculate Points for Function \( f(x) + g(x) \)

Add the \( y \) values of \( f(x) \) and \( g(x) \) for each selected value of \( x \). Example: \(-2: f(-2) + g(-2) = 4 - \frac{8}{3} = \frac{4}{3}\), \(-1: f(-1) + g(-1) = 1 - \frac{1}{3} = \frac{2}{3}\), \(0: f(0) + g(0) = 0\), \(1: f(1) + g(1) = 1 + \frac{1}{3} = \frac{4}{3}\), \(2: f(2) + g(2) = 4 + \frac{8}{3} = \frac{20}{3}\). Plot these points and connect them to form the graph of \( f(x) + g(x) \).

Step 5: Graphing All Functions on a Common Screen

Draw all three graphs on the same set of axes. You will observe the parabola \( f(x) \), the cubic curve \( g(x) \), and their sum \( f(x) + g(x) \) intersecting or merging at certain points. This visually illustrates the concept of graphical addition.

Key Concepts

Quadratic Function

A quadratic function is one of the simplest types of polynomial functions and is often represented in the standard form as \( f(x) = ax^2 + bx + c \). For our specific function, \( f(x) = x^2 \), it's a special case where \( a = 1 \), \( b = 0 \), and \( c = 0 \). This function graphs as a parabola opening upwards.

The vertex of this parabola is at the origin point \((0,0)\), which is also its minimum point since the parabola opens upward. The axis of symmetry is the vertical line \( x = 0 \). This symmetry means that the graph is a mirror image on either side of this line.

When plotting a quadratic function like \( x^2 \), every \( x \) value equidistant from the axis of symmetry will have the same \( y \) value, which forms the characteristic U-shape of a parabola:

• At \( x = 0 \): \( f(0) = 0 \)
• At \( x = 1 \) and \( x = -1 \): \( f(x) = 1 \)
• At \( x = 2 \) and \( x = -2 \): \( f(x) = 4 \)

As the absolute value of \( x \) increases, \( y \) increases much more rapidly due to the square — this is why quadratic functions grow parabolically.

Cubic Function

A cubic function is a polynomial of degree three, typically expressed as \( g(x) = ax^3 + bx^2 + cx + d \). In our problem, the specific cubic function is \( g(x) = \frac{1}{3}x^3 \), where \( a = \frac{1}{3} \) and all other coefficients \( b, c, \) and \( d \) are zero.

The graph of this function is an S-shaped curve that passes through the origin, as there is no constant term to shift it up or down. This type of curve is symmetric about the origin, meaning that if you rotate it 180 degrees, it appears the same. This characteristic is called rotational symmetry.

Let's consider a few key points for \( g(x) = \frac{1}{3}x^3 \):

• At \( x = 0 \): \( g(0) = 0 \)
• At \( x = 1 \): \( g(1) = \frac{1}{3} \)
• At \( x = -1 \): \( g(-1) = -\frac{1}{3} \)
• At \( x = 2 \): \( g(2) = \frac{8}{3} \)
• At \( x = -2 \): \( g(-2) = -\frac{8}{3} \)

Cubic functions like \( \frac{1}{3}x^3 \) continue to glide both upwards and downwards far from the origin, giving them a distinctive S-curve appearance.

Graphical Addition

Graphical addition involves generating a new graph representing the sum of the \( y \)-values of existing functions at each point \( x \). It's a visual understanding of addition in algebra, where the values of the functions are added together at each point to create a new curve.

For our functions, \( f(x) = x^2 \) (quadratic) and \( g(x) = \frac{1}{3}x^3 \) (cubic), their combined function is \( h(x) = f(x) + g(x) = x^2 + \frac{1}{3}x^3 \). At every chosen input \( x \), the graphs of these functions provide points:

• \( x = -2 \): \( h(-2) = 4 - \frac{8}{3} = \frac{4}{3} \)
• \( x = -1 \): \( h(-1) = 1 - \frac{1}{3} = \frac{2}{3} \)
• \( x = 0 \): \( h(0) = 0 \)
• \( x = 1 \): \( h(1) = 1 + \frac{1}{3} = \frac{4}{3} \)
• \( x = 2 \): \( h(2) = 4 + \frac{8}{3} = \frac{20}{3} \)

These computed points can be plotted on a graph to reveal the shape of this new function, \( h(x) \).

Placing all three graphs — the two original functions and their sum — on one set of axes captures how they coincide or diverge, illustrating the influence of each function on the resultant graph.