Question

1. what vertical translation is applied to \\(y = x^2\\) if the transformed graph passes through the point \\((4, 19)\\)?

1. what horizontal translation is applied to \\(y = x^2\\) if the translation image graph passes through the point \\((5, 16)\\)?

Explanation:

Analyze vertical translation

Using the Vertical Translation knowledge point.
A vertical translation of \(y = x^2\) is represented by:

$$y = x^2 + k$$

We substitute the given point \((4, 19)\) into this equation to solve for \(k\):

$$19 = 4^2 + k$$

$$19 = 16 + k$$

$$k = 3$$

Since \(k = 3\), the graph is translated 3 units up.

Analyze horizontal translation

Using the Horizontal Translation knowledge point.
A horizontal translation of \(y = x^2\) is represented by:

$$y = (x - h)^2$$

We substitute the given point \((5, 16)\) into this equation to solve for \(h\):

$$16 = (5 - h)^2$$

Taking the square root of both sides gives two possible cases:

$$5 - h = 4 \implies h = 1$$

$$5 - h = -4 \implies h = 9$$

Thus, the translation can be 1 unit to the right or 9 units to the right.

Answer:

Question 6

A vertical translation of 3 units up.

Question 7

A horizontal translation of 1 unit to the right or 9 units to the right.