Consider the equation y=-2√3x-12 -5
a. Write the equation in function notation, y=af(k(x-c))+d.
b. Describe the transformations that have taken place from base the function.
I assume you mean
y = -2√(3x-12)-5
= -2√(3(x-4)) - 5
so, the function √x has been shifted right by 4, scaled horizontally by a factor of 1/3, reflected over the x-axis, stretched vertically by a factor of 2, and shifted down 5.
To write the given equation in function notation, y = af(k(x - c)) + d, we need to identify the values of a, f, k, c, and d.
Let's start by looking at the given equation: y = -2√3x - 12 - 5.
a. To find the value of a, we need to determine the coefficient multiplying the function. In this case, there is no coefficient in front of the square root term. Therefore, a = 1.
Next, let's analyze the square root term: -2√3x. This can be rewritten as -2 * (√3) * x. The term inside the square root, √3, represents the base function. So f = √3.
Now, let's consider the constant term, -12 - 5 = -17. This value represents the vertical shift of the graph, which is represented by the term d. Therefore, d = -17.
To identify the value of k and c, we need to manipulate the equation in the form of y = af(k(x - c)) + d.
The given equation, y = -2√3x - 12 - 5, can be rearranged as follows:
y = √3 * (-2x) - 17.
Comparing this expression to the function notation, we can deduce that k = -2 and c = 0.
Therefore, the equation y = -2√3x - 12 - 5 can be written in function notation as y = √3 * (-2(x - 0)) - 17.
b. Now, let's describe the transformations that have taken place from the base function, y = √3x.
1. Reflection: The negative sign in front of 2 (√3) causes a reflection across the x-axis. This means that the graph of the function is flipped vertically.
2. Horizontal stretch/compression: The value of k = -2 indicates a horizontal stretch/compression. In this case, since k = -2, the graph is compressed horizontally by a factor of two. The graph is narrower compared to the base function.
3. Horizontal shift: The value of c = 0 implies no horizontal shift. The graph is centered on the y-axis.
4. Vertical shift: The constant term, d = -17, represents a vertical shift of -17 units. The graph is shifted downward by 17 units compared to the base function.
In summary, the given function represents a vertically flipped, horizontally compressed (by a factor of two), centered on the y-axis, and shifted downward by 17 units compared to the base function y = √3x.