Write an equation for the function graphed below. The y intercept is at (0,-0.2) 5+ 4 3- 2- 1- -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 -1- -2- -3- -4- -5+ Q y =
The Correct Answer and Explanation is:
Here is the equation for the function graphed:
y = -0.8(x – 1) / (x² – 4)
Explanation:
To find the equation of the graphed rational function, we can identify its key features from the image and use them to construct the equation step-by-step.
- Vertical Asymptotes: The graph shows two vertical dashed lines, which are the vertical asymptotes. These occur at x = -2 and x = 2. Vertical asymptotes correspond to the roots of the denominator. Therefore, the denominator of our function will have factors of (x + 2) and (x – 2). Multiplying these gives the denominator: (x + 2)(x – 2) = x² – 4.
- Horizontal Asymptote: The graph approaches the x-axis (the line y = 0) as x goes to positive and negative infinity. This means the horizontal asymptote is y = 0. For a rational function, this occurs when the degree of the numerator is less than the degree of the denominator. Our denominator has a degree of 2, so this condition is met.
- X-intercept: The graph crosses the x-axis at x = 1. The x-intercepts of a rational function correspond to the roots of the numerator. This means the numerator has a factor of (x – 1).
- General Equation: Combining these features, we can write a general form for the function:
y = a * (x – 1) / (x² – 4)
Here, ‘a’ is a constant scaling factor that we need to determine. - Finding the Constant ‘a’: The problem states that the y-intercept is at (0, -0.2). We can use this point to solve for ‘a’ by substituting x=0 and y=-0.2 into our general equation:
-0.2 = a * (0 – 1) / (0² – 4)
-0.2 = a * (-1) / (-4)
-0.2 = a / 4To isolate ‘a’, we multiply both sides by 4:
a = 4 * (-0.2)
a = -0.8 - Final Equation: Now we substitute the value of a = -0.8 back into the general equation to get the final answer:
y = -0.8(x – 1) / (x² – 4)
This equation can also be written as y = (-0.8x + 0.8) / (x² – 4)
