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Chapter 7: Problem 21

(Graphing program required.) Determine which of the graphs of the following pairs of functions intersect. If the graphs intersect, find the point or points of intersection. a. \(y=2 x \quad y=4 x^{2}\) d. \(y=x^{-1} \quad y=x^{-2}\) b. \(y=4 x^{2} \quad y=4 x^{3}\) e. \(y=4 x^{-2} \quad y=4 x^{-3}\) c. \(y=x^{-2} \quad y=4 x^{2}\)

Short Answer

Expert verified
a. (0, 0) and (0.5, 1); b. (0, 0) and (1, 4); c: no intersection; d: (1, 1); e: (1, 4).

Step by step solution

01

Set the Equations Equal

For each pair of functions, set the equations equal to each other to find the points of intersection.
02

Solve the Equation for Each Pair

Solve the resulting equation for the variable x to find the x-coordinates where the intersections might occur.
03

Verify the Points of Intersection

Plug the x-coordinates back into the original functions to find the corresponding y-coordinates and ensure they match for each pair.
04

Example a: Set the Equations Equal

For the first pair, set the equations equal: \( 2x = 4x^2 \) Solve for x. \( 0 = 4x^2 - 2x \) \( 0 = 2x(2x - 1) \) x = 0 or x = 0.5
05

Example a: Verify the Points of Intersection

Plug x = 0 into the original equations: \( y = 2(0) = 0 \) \( y = 4(0)^2 = 0 \) Plug x = 0.5 into the original equations: \( y = 2(0.5) = 1 \) \( y = 4(0.5)^2 = 1 \) The graphs intersect at (0, 0) and (0.5, 1).
06

Example b: Set the Equations Equal & Solve

Set: \( 4x^2 = 4x^3 \) Simplify: \( x^2 = x^3 \) \( x^2(1 - x) = 0 \) x = 0 or x = 1
07

Example b: Verify the Points of Intersection

For x = 0: \( y = 4(0)^2 = 0 \) \( y = 4(0)^3 = 0 \) For x = 1: \( y = 4(1)^2 = 4 \) \( y = 4(1)^3 = 4 \) The graphs intersect at (0, 0) and (1, 4).
08

Example c: Set the Equations Equal & Solve

Set: \( x^{-2} = 4x^2 \) \( 1/x^2 = 4x^2 \) \( 1 = 4x^4 \) x = \( \frac{1}{\root{4}{4}} \) and \( -\frac{1}{\root{4}{4}} \) x = \( \frac{1}{2} \) and \( -\frac{1}{2} \)
09

Example c: Verify the Points of Intersection

For x = \( \frac{1}{2} \): \( y = \left(\frac{1}{2}\right)^{-2} = 4 \) \( y = 4\left(\frac{1}{2}\right)^2 = 1 \) For x = \( -\frac{1}{2} \): Same calculation for y-values, yielding y = 4 and 1 No intersection due to different y-values at the same x-values.
10

Example d: Set the Equations Equal & Solve

Set: \( x^{-1} = x^{-2} \) \( 1/x = 1/x^2 \) Multiply both sides by x^2: x = 1
11

Example d: Verify the Points of Intersection

For x = 1: \( y = 1^{-1} = 1 \) \( y = 1^{-2} = 1 \) The graphs intersect at (1, 1).
12

Example e: Set the Equations Equal & Solve

Set: \( 4x^{-2} = 4x^{-3} \) Simplify: x = 1
13

Example e: Verify the Points of Intersection

For x = 1: \( y = 4/1^2 = 4 \) \( y = 4/1^3 = 4 \) The graphs intersect at (1, 4).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Methods
When trying to find the intersection points between graphs of functions, algebraic methods provide a useful tool. Start by setting the equations of the functions equal to each other. This approach will help you determine the x-coordinates where the functions potentially intersect. For instance, if you have the equations: \(y = 2x\) and \(y = 4x^2\), set them equal: \(2x = 4x^2\). This step converts the problem into solving an algebraic equation. Break down this equation into simpler parts by factoring or using other algebraic techniques, giving you possible x-values.
Function Intersection
Understanding function intersection is fundamental in graphing and solutions. Intersection points occur where the y-values of the functions are the same for the same x-values. Start with solving the equations to find common x-values by equating the two functions. Next, plug the x-values back into the original functions to see where the y-values are equal. For example, if the x-values from our previous step are 0 and \((0.5)\), check the y-values for both functions at these points. If both functions return the same y-value at these x-points, such as \((2(0) = 0)\) and \((4(0)^2 = 0)\), then the intersection points are at these coordinates.
Equation Solving
Solving the equation derived from setting the functions equal is a crucial step. Use algebraic techniques such as factoring, quadratic formula, or root-solving strategies. For instance, in the equation \(4x^2 - 2x = 0\), factor out common terms to get \(2x(2x - 1) = 0\), leading to solutions \(x = 0\) or \(x = 0.5\). This process gives possible x-values. Once you have these x-values, substitute them back into the original functions to verify the corresponding y-values are equal. Equation solving breaks down complex problems into manageable parts.
Graph Verification
After finding the potential intersection points, always verify them graphically. Plot both functions on the same graph and look for points where they cross. This step confirms the algebraic solution visually. For example, if the functions are \(y = x^{-1}\) and \(y = x^{-2}\), after setting the equations and solving x^{-1} = x^{-2}\, you find \(x = 1\). Verify this by plotting and checking if both functions intersect at \((1, 1)\). Verifying graphically ensures there are no calculation mistakes and solidifies understanding by seeing the actual intersection.

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