In the xy-plane, the parabola defined by y = 2x^2 - 8x + 5 is reflected across the line y … — GRE General Test

In the $xy$ -plane, the parabola defined by $y = 2x^2 - 8x + 5$ is reflected across the line $y = x$ . The resulting curve is then translated $3$ units to the right and $4$ units downward to produce curve $C$ . If the point $(a, -1)$ lies on curve $C$ , what is the value of $a$ ?

Answer: 2

Answer

The value of

a

2

Completing the square yields

y = 2(x - 2)^2 - 3

. Reflecting across

y = x

interchanges

x

and

y

, producing

x = 2(y - 2)^2 - 3

. Shifting this curve

3

units to the right replaces

x

with

x - 3

, and shifting

4

units down replaces

y

with

y + 4

. The equation simplifies to

x = 2(y + 2)^2

. Setting

y = -1

gives

a = 2(-1 + 2)^2 = 2

Step-by-Step Solution

Rewrite the original parabola equation in vertex form.

y = 2(x - 2)^2 - 3

Expressing the parabola in vertex form makes tracking coordinate transformations straightforward.

Apply the reflection across the line

y = x

x = 2(y - 2)^2 - 3

Reflecting across

y = x

interchanges the

x

and

y

coordinates of every point on the graph.

Apply the translations to find the equation of curve

C

x = 2(y + 2)^2

Translating right by

3

replaces

x

with

x - 3

, and translating down by

4

replaces

y

with

y + 4

Substitute

y = -1

into the equation for curve

C

to calculate

a

a = 2

Since the point

(a, -1)

lies on curve

C

, substituting

x = a

and

y = -1

yields

a = 2(1)^2 = 2

Key Concept

Reflection across

y = x

and rigid horizontal/vertical translations of conic sections

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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