Problem 14 - Algebra and Trigonometry

Question

\(13-16\) Draw the graphs of \(f, g,\) and \(f+g\) on a common screen to illustrate graphical addition. $$ f(x)=x^{2}, \quad g(x)=\sqrt{x} $$

Step-by-Step Solution

Answer

Plot functions \(f(x)=x^2\), \(g(x)=\sqrt{x}\), and \(f+g"). Notice that \(f+g\) is above both \(f\) and \(g\) on all non-negative \(x\).

Step 1: Understand Each Function Individually

Let's first understand the functions individually. The function \( f(x) = x^2 \) is a parabola that opens upwards and passes through the origin \((0,0)\). As \( x \) increases or decreases from the origin, the value of \( f(x) \) increases.

Step 2: Explore the Second Function

The function \( g(x) = \sqrt{x} \) represents the square root function. It starts at the origin \((0,0)\), and as \( x \) increases, \( g(x) \) increases but at a decreasing rate. This creates a curve that rises slowly, staying above the x-axis for non-negative values of \( x \).

Step 3: Determine \( f(x) + g(x) \)

The function \( f(x) + g(x) = x^2 + \sqrt{x} \) combines the effects of the parabola and the square root functions. For each value of \( x \), calculate \( f(x) \) and \( g(x) \) separately and then add them to get \( f(x) + g(x) \).

Step 4: Graph Each Function

On the same set of axes, plot the graphs of \( f(x) = x^2 \), \( g(x) = \sqrt{x} \), and \( f(x) + g(x) = x^2 + \sqrt{x} \). Use different colors or line styles for each function to distinguish among them.

Step 5: Analyze the Graphical Addition

Observe how \( f + g \) behaves for different values of \( x \). Notice that \( f + g \) is above both \( f \) and \( g \) for all non-negative \( x \) since we are adding the values of two non-negative functions.

Key Concepts

Parabola

A parabola is a symmetrical curve that you will often encounter in algebra and calculus. It is the graphical representation of a quadratic function, such as \( f(x) = x^2 \). This kind of function displays a characteristic U-shape when graphed on a coordinate plane. The lowest point of this curve is called the vertex, and in our case with \( f(x) = x^2 \), the vertex is at the origin, or point \((0,0)\).

The parabola opens upwards because the coefficient of \( x^2 \) is positive. Imagining the graph, as the value of \( x \) increases or decreases from zero, \( f(x) \) rapidly grows since squaring a number, whether positive or negative, results in a positive value. Key features of parabolas include:

• A vertex that marks the parabola's minimum or maximum point
• An axis of symmetry, which is a vertical line that runs through the vertex
• The overall shape resembles a U for upward-opening parabolas

Understanding how parabolas function will help you see how they interact with other functions on a graph.

Square Root Function

The square root function is graphically represented by \( g(x) = \sqrt{x} \). This function is defined only for non-negative values of \( x \) because you can't take the square root of a negative number in real numbers. At \( x = 0 \), its value is \( g(0) = 0 \), so it starts at the origin.

The graph of the square root function looks like a gradually ascending curve. As \( x \) increases, the rate at which \( g(x) \) grows diminishes, which is why the curve levels out rather than shooting up rapidly like a parabola.

• The start point is at the origin \((0,0)\)
• It only rises on non-negative x-values
• The slope of the curve decreases as \( x \) gets larger

This function is crucial in understanding how different mathematical operations that involve roots behave when plotted. It shows how slow-increasing functions look on a graph.

Function Graphing

Graphing functions allows you to visually interpret equations and understand the relationship between variables. When graphing multiple functions on the same set of axes, like \( f(x) = x^2 \) and \( g(x) = \sqrt{x} \), you can explore how they interact through graphical addition.

By plotting \( f(x) \) and \( g(x) \) separately, then finding \( f(x) + g(x) \), you're performing a key concept called "graphical addition". This involves adding the values of \( f(x) \) and \( g(x) \) for each \( x \) and plotting the result.

• Plot each function separately
• Use distinct colors or line styles to differentiate them
• Combine their effects by adding their y-values for each x

Graphical addition helps in seeing how two separate mathematical functions combine to form a new function, enabling a clearer comprehension of composite functions and their characteristics on a coordinate plane.