Question

Please answer and explain your response in depth. Thank you in advance.

Answer 1

The parent function is given below as

`y=x^2`

Step 1:

For the function to be shifted 2 units to the left, we will have the equation below

The function h(x) = f(x + a) represents a horizontal shift a units to the left. Informally: Adding a positive number after the x inside the parentheses shifts the graph left,

`y=(x+2)^2`

The graph is represented below as

Step 2:

The function above is then shifted 10 units down, this can be represented with the function below

When a function shifts vertically, the y-value changes. The x-value stays the same, while the y-value changes the amount of the shift. If it shifts up, then we add the value to the y-term. If it shifts down, we will subtract that value from the y-term.

`y=(x+2)^2-10`

The function is represented below as

Step 3:

The function above is then flipped to open downwards

Reflections. Parabolas can also be reflected in the x-axis. Thus the parabola y=−x2 is a reflection of the basic parabola in the x-axis.

Hence,

The function will be

`y=-(x+2)^2-10`

Hence,

The graph is given below as

Therefore,

The function that defines the curve will be given below as

`y=-\mleft(x+2\mright)^2-10`

The graph is given below as

Hence,

The final answer is y = -(x+2)²- 10