Problem 46 - Algebra 2

Question

Graph each pair of functions. Find the approximate point(s) of intersection. \(y=\frac{4}{x^{2}}+2, y=9\)

Step-by-Step Solution

Answer

The functions \(y=\frac{4}{x^{2}}+2\) and \(y=9\) intersect at the points \((- \sqrt{\frac{4}{7}},9)\) and \((\sqrt{\frac{4}{7}},9\).

Step 1: Graph the first function

The first function in this exercise is \(y=\frac{4}{x^{2}}+2\). This function is a quadratic function shifted upwards by 2. To graph this, first graph the function \(y=\frac{4}{x^{2}}\) and then shift this graph two units upwards. This function has a center at the origin and opens upwards and downwards.

Step 2: Graph the second function

The second function in this exercise is \(y=9\). This function represents a horizontal line at \(y=9\). This line will intersect the graph from Step 1 at the points of intersection we are trying to find.

Step 3: Find the points of intersection

The points of intersection are where the two graphs from steps 1 and 2 meet. To mathematically find these points, set \(y=\frac{4}{x^{2}}+2\) equal to \(y=9\) and solve for \(x\). This yields the equation \(\frac{4}{x^{2}}+2=9\). Solving for \(x\) gives \(x=\pm\sqrt{\frac{4}{7}}\). Therefore, the points of intersection are at \((- \sqrt{\frac{4}{7}},9)\) and \( (\sqrt{\frac{4}{7}},9)\)

Key Concepts

Graphing Functions

The process of graphing functions entails creating a visual representation of each mathematical equation on a coordinate system. Graphing helps in understanding how a function behaves and determines its key features. Let's explore this process using the functions given in the exercise.

First, consider the function \( y = \frac{4}{x^2} + 2 \). This is a quadratic function that forms part of a type of function known as hyperbolas. To graph it:

• Begin by graphing \( y = \frac{4}{x^2} \). This base function forms a curve that approaches the x-axis but never touches it due to its asymptotic nature, meaning the function’s values grow infinitely as \( x \) approaches zero.
• After plotting \( y = \frac{4}{x^2} \), shift the graph upwards by 2 units to account for the \(+2\) in the original function \( y = \frac{4}{x^2} + 2 \). This shift changes the horizontal asymptote from \( y = 0 \) to \( y = 2 \).

Next, graph the second function, \( y = 9 \), which is straightforward:

• This is a horizontal line intersecting the y-axis at 9. It will appear as a straight line running parallel to the x-axis at the point \( y = 9 \).

By plotting these functions on the same graph, you can visually assess where they intersect. This visual approach confirms the mathematical solutions which we'll explore next.

Quadratic Functions

Quadratic functions, a cornerstone of algebra, often take the form \( f(x) = ax^2 + bx + c \). They typically graph as parabolas, but can also manifest in other forms depending on transformations applied. In the exercise at hand, we are dealing with the function \( y = \frac{4}{x^2} + 2 \), a type of rational function resembling an inverted parabola.

The term "quadratic" in this context refers to the power of \( x \), specifically \( x^2 \). In the function \( y = \frac{4}{x^2} \), the quadratic aspect comes from \( x^2 \) being in the denominator, giving it a unique hyperbolic shape rather than the typical U-shape of standard parabolas.

Key features of quadratic functions to consider include:

• Vertex: The highest or lowest point of a parabola; in our function, the curve rudiments can see vertex-like behavior as it opens vertically upward from a line.
• Symmetry: Quadratic functions are symmetric around their vertex or axis of symmetry. For our function, symmetry is still apparent as symmetry lines appear vertically through the origin.
• Asymptotes: As our function features a hyperbolic curve, it contains asymptotes, lines that the curve comes close to but never touches. Here, that’s the x-axis \( y = 0 \).

Understanding these characteristics aids in grasping the function’s behavior and accurately predicting its graphical portrayal.

Solving Equations

Finding the intersection point(s) of two functions typically involves solving an equation set to find where \( y \)-values are equal for given \( x \)-values. In this exercise, this means setting \( y = \frac{4}{x^2} + 2 \) equal to \( y = 9 \).

Follow these steps to solve the equation:

• Start by setting the equations equal: \( \frac{4}{x^2} + 2 = 9 \).
• Isolate \( x^2 \) by first subtracting 2 from both sides, resulting in \( \frac{4}{x^2} = 7 \).
• Next, multiply both sides by \( x^2 \) and divide by 7 to form the equation \( x^2 = \frac{4}{7} \).
• Take the square root of both sides to solve for \( x \), resulting in \( x = \pm \sqrt{\frac{4}{7}} \).

This solution gives you the x-coordinates of the points where the graphs intersect, at \( x = \sqrt{\frac{4}{7}} \) and \( x = -\sqrt{\frac{4}{7}} \). Correspondingly, these occur at the y-coordinate from the line \( y = 9 \), forming the points of intersection \((-\sqrt{\frac{4}{7}}, 9)\) and \((\sqrt{\frac{4}{7}}, 9)\). Understanding this procedure allows you to determine intersection points for various functions beyond this example.