Exercises: Identify transformations of graphs - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

If $f(x) = x^2$ , what is $f(3)$ ?

A.

6

B.

9

C.

5

D.

32

2.

The graph of $f(x) = x^2$ passes through the point $(2, 4)$ . If every $y$ -value is increased by 5, what is the new point?

A.

$(7, 4)$

B.

$(2, 9)$

C.

$(7, 9)$

D.

$(2, -1)$

3.

A graph is symmetric about the $y$ -axis. If the point $(3, 7)$ is on the graph, which other point must also be on the graph?

A.

$(-3, 7)$

B.

$(3, -7)$

C.

$(-3, -7)$

D.

$(7, 3)$

B

Fluency Practice

Apply the transformation rules to identify the effect on the graph.

C

Varied Practice

Apply your understanding of transformations in different formats.

D

Word Problems

Use your knowledge of transformations to solve each problem.

1.

The graph of a function $f$ has its vertex at $(0, 0)$ . A transformed version of the graph has its vertex at $(0, -6)$ . The transformation is of the form $y = f(x) + k$ .

What is the value of $k$ ?

2.

A parabola $y = f(x)$ has its vertex at $(-1, 3)$ . After a horizontal translation of the form $y = f(x - h)$ , the vertex is at $(4, 3)$ .

What is the value of $h$ ?

3.

A scientist models a population with $P(t) = t^2$ . She adjusts the model to $Q(t) = P(t - 4) + 10$ .

1.

Which describes the transformation from $P$ to $Q$ ?

A.

Shift right 4 units and up 10 units

B.

Shift left 4 units and up 10 units

C.

Shift right 4 units and down 10 units

D.

Shift left 4 units and down 10 units

2.

If $P(0) = 0$ , what is $Q(4)$ ?

E

Error Analysis

Each problem shows student work that contains an error. Identify the mistake.

1.

Taylor says: "The graph of $y = f(x + 5)$ is the graph of $y = f(x)$ shifted 5 units to the right."

What is wrong with Taylor's statement?

A.

The shift is to the left, not the right. Adding 5 inside the function shifts the graph in the opposite direction from the sign.

B.

The shift is vertical (up 5), not horizontal.

C.

Both the $x$ - and $y$ -coordinates shift by 5.

D.

There is no error; $f(x + 5)$ shifts right 5.

2.

Priya claims: " $f(x) = x^3 + 1$ is an odd function because $x^3$ is odd."

What is the error in Priya's reasoning?

A.

$f(-x) = -x^3 + 1$ , which is neither $f(x)$ nor $-f(x)$ , so the function is neither even nor odd. The constant term $+1$ breaks the odd symmetry.

B.

$x^3 + 1$ is actually an even function because 1 is an even number.

C.

You cannot classify functions with mixed terms as even or odd.

D.

$f(x) = x^3 + 1$ is odd because all cubic functions are odd.

F

Challenge

These problems require deeper reasoning about transformations.

1.

A student applies transformations to $y = x^2$ in this order: shift right 3, then stretch vertically by 2, then shift up 4. Write the resulting equation. Then explain whether changing the order to "stretch vertically by 2, shift up 4, shift right 3" gives the same result.

2.

Let $f(x) = x^4 - 2x^2$ . Is $f$ even, odd, or neither? Choose the answer with correct justification.

A.

Even, because $f(-x) = (-x)^4 - 2(-x)^2 = x^4 - 2x^2 = f(x)$

B.

Odd, because the highest power is 4 and the function has a symmetric shape

C.

Neither, because the function has both $x^4$ and $x^2$ terms

D.