Write the following quadratic function in vertex form. Write the mapping notation and describe any transformations of the graph. 𝑦 =1/4 𝑥^2 − 4𝑥 + 5
To write the quadratic function 𝑦 = 1/4𝑥^2 − 4𝑥 + 5 in vertex form, we need to complete the square.
First, let's factor out the common factor of 1/4 from the first two terms:
𝑦 = 1/4(𝑥^2 − 16𝑥) + 5
Next, we need to complete the square by adding and subtracting a constant term inside the parentheses that will allow us to create a perfect square trinomial. In this case, we need to add (−4/2)^2 = 4 to complete the square:
𝑦 = 1/4(𝑥^2 − 16𝑥 + 4 − 4) + 5
Now, we can rewrite the expression inside the parentheses as a perfect square trinomial:
𝑦 = 1/4((𝑥 − 4)^2 − 4) + 5
Expanding and simplifying the expression, we have:
𝑦 = 1/4(𝑥 − 4)^2 − 1 + 5
𝑦 = 1/4(𝑥 − 4)^2 + 4
Therefore, the quadratic function 𝑦 = 1/4𝑥^2 − 4𝑥 + 5 in vertex form is 𝑦 = 1/4(𝑥 − 4)^2 + 4.
The mapping notation for this quadratic function is: f(x) → 1/4(x − 4)^2 + 4.
The graph of this quadratic function has undergone several transformations compared to the standard quadratic function y = 𝑥^2.
1. Vertical stretch: The coefficient 1/4 stretches the graph vertically by making it flatter and wider compared to the standard quadratic function.
2. Vertical translation: The constant term 4 shifts the graph upward by 4 units. The vertex of the parabola is now located at (4, 4).
3. Horizontal translation: The term (x - 4) shifts the graph 4 units to the right compared to the standard quadratic function.
Therefore, comparing to the standard quadratic function y = 𝑥^2, the graph of the quadratic function 𝑦 = 1/4𝑥^2 − 4𝑥 + 5 has been vertically stretched, upwardly translated, and horizontally translated.