In the xy-plane, the graph of y = x^2 - 8x + 12 represents the quadratic function f. If th… — SAT

In the $xy$ -plane, the graph of $y = x^2 - 8x + 12$ represents the quadratic function $f$ . If this graph is shifted $3$ units to the left and $5$ units up to create the graph of a new function $g$ , what is the $y$ -value of the vertex of the graph of $g$ ?

Answer: 1

Answer

The y-value of the vertex of the graph of g is 1.

The vertex of the original quadratic function f(x) = x^2 - 8x + 12 can be found by rewriting it in vertex form, which is f(x) = (x - 4)^2 - 4. This shows that the vertex of the original graph is (4, -4). A translation of 3 units to the left subtracts 3 from the x-coordinate of the vertex (4 - 3 = 1), and a translation of 5 units up adds 5 to the y-coordinate of the vertex (-4 + 5 = 1). Thus, the vertex of the graph of the new function g is (1, 1), making its y-value 1.

Step-by-Step Solution

Find the vertex of the original quadratic function.

The vertex of the original graph is

(4, -4)

By completing the square on

y = x^2 - 8x + 12

, we get

y = (x - 4)^2 - 4

, which reveals the vertex is at

(4, -4)

Apply the translation to the vertex coordinates.

The translated vertex coordinates are

(1, 1)

Shifting a point

(x, y)

3

units left and

5

units up results in the point

(x - 3, y + 5)

. Applying this to the vertex

(4, -4)

yields

(4 - 3, -4 + 5) = (1, 1)

Identify the y-coordinate of the new vertex.

The y-value is

1

The vertex of the graph of

g

(1, 1)

, where the second coordinate represents the y-value.

Key Concept

Identifying the vertex of a quadratic function and applying horizontal and vertical translations in the coordinate plane.

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