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Magazine Chapter 14: Problem 28
Solve the given systems of equations graphically by using a calculator. Find all values to at least the nearest 0.1. $$\begin{aligned} &y=\cos x\\\ &y=\log _{3} x \end{aligned}$$
Short Answer
Expert verified
The intersection points are approximately \((0.9, 0.5)\) and \((3.5, -0.9)\).
Step by step solution
01
Understand the Equations
We have two equations: the first is a trigonometric function, \( y = \cos x \), and the second is a logarithmic function, \( y = \log_3 x \). To solve graphically, we'll sketch both graphs and find their intersection points.
02
Set Up a Graphing Calculator
Enter both equations into a graphing calculator. Set the calculator to radians for the trigonometric function and adjust the viewing window to ensure visibility of possible intersections, likely \(x \in [0, 5]\) and \(y \in [-1, 1]\).
03
Graph both Functions
Plot the graph of \( y = \cos x \), which oscillates between -1 and 1 periodically. Next, plot \( y = \log_3 x \), which starts from negative infinity as \(x\) approaches 0 from the positive side and increases gradually.
04
Identify Intersection Points
Look for the points where the graphs of \( y = \cos x \) and \( y = \log_3 x \) intersect. Use the calculator's intersection or tracing feature to find the approximate \(x\)-values at the intersection points to the nearest 0.1.
05
Verify the Intersections
Determine the corresponding \(y\)-values for each intersection \(x\)-value by substituting \(x\) back into either equation (both should yield the same \(y\)). Ensure both values are accurate by comparing against the equations directly.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Functions
Trigonometric functions are fundamental in mathematics and have a periodic nature. In our problem, we work with the cosine function, denoted as \( y = \cos x \). This function outputs results between -1 and 1, capturing the oscillating motion of waves. Here are some key characteristics to understand the cosine function better:
- Amplitude: The maximum height from the center line of the wave to its peak, for cosine, it's always 1.
- Period: This tells us how long it takes for the function to repeat its pattern. For cosine, the period is \(2\pi\), translating to approximately 6.2832 radians, which means it completes a full cycle in that length.
- Phase Shift: Cosine waves can shift horizontally. However, the basic formula used here does not include a shift, starting at its max point at \( x = 0 \).
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions. In this exercise, we have the specific function \( y = \log_3 x \). Several unique aspects of logarithmic functions are useful here:
- Domain: The function is only defined for positive \(x\) values, emphasizing that the graph begins at \( x > 0 \).
- Vertical Asymptote: As \( x \) approaches zero from the positive side, the function tends toward negative infinity, meaning the graph never touches the y-axis.
- Base: The base here is 3, indicating the rate of growth as \( x \) varies, which is slower compared to logs of higher bases like 10.
- Intersection with the axes: Since logs are undefined for zero or negative \( x \), they only intersect the positive \( x \)-axis when \( y = 0 \), and \( x = 1 \).
Graphing Calculator
A graphing calculator is a powerful tool for visualizing mathematical functions. For the given equations, setting up the calculator correctly is essential:
- Mode Setting: Ensure the mode is set to radians, which is common for trigonometric functions, as they typically use the radian measure.
- Entering Functions: Input both equations manually; \( y = \cos x \) and \( y = \log_3 x \).
- Window Adjustment: Choose a viewing window to capture both functions completely and allow intersections, like from 0 to 5 for \(x\) and -1 to 1 for \(y\).
- Graph Plotting: By plotting, you'll observe oscillating behavior of the cosine function and the gradually increasing log function.
Intersection Points
Intersection points are locations on a graph where two or more functions have the same values for \(x\) and \(y\). To find these points:
- Graph Observation: Watch where the curves of \( y = \cos x \) and \( y = \log_3 x \) intersect visually on the graphing calculator.
- Use Calculator Features: Utilize the calculator's tracing or intersection tool to get precise \(x\)-values where the intersections occur.
- Verify Intersection: Once the \(x\)-value is determined, substitute it back into either equation to ensure the same \(y\)-value appears, confirming an intersection.
- Approximation: Keep values to at least one decimal place for precision, which is often sufficient for most practical exercises.
