Question

1. use the graph of each transformed parent function to describe the transformation and write its equation.

a.

transformation(s):
reflect over x

$f(x) = $ _______________

b.

transformation(s):

$f(x) = $ _______________

c.

transformation(s):

$f(x) = $ _______________

d.

transformation(s):

$f(x) = $ _______________

Explanation:

Part a:

Step 1: Identify Parent Function

The parent function for a parabola opening upwards is \( y = x^2 \). The graph in part a is a parabola opening upwards, shifted down by 1 unit (vertex at (0, -1)).

Step 2: Write the Equation

The transformation is a vertical shift down by 1. So the equation is \( f(x)=x^2 - 1 \).

Part b:

Step 1: Identify Parent Function

Parent function \( y = x^2 \). The graph is shifted left (since vertex is at (-3, 0)) and maybe reflected? Wait, no, the vertex is at (-3, 0), opening upwards. Wait, the vertex form of a parabola is \( y = a(x - h)^2 + k \), where (h,k) is vertex. Here vertex is (-3, 0), so \( h=-3, k = 0 \), \( a = 1 \) (since it's same width as parent). So transformation: horizontal shift left by 3 units.

Step 2: Write the Equation

So \( f(x)=(x + 3)^2 \) (since \( y = (x - (-3))^2 + 0 \)).

Part c:

Step 1: Identify Parent Function

Parent function \( y = x^2 \). The graph has vertex at (3, -3), opening upwards. So transformation: horizontal shift right by 3, vertical shift down by 3.

Step 2: Write the Equation

Using vertex form \( y = (x - 3)^2 - 3 \), so \( f(x)=(x - 3)^2 - 3 \).

Part d:

Step 1: Identify Parent Function

Parent function is \( y = -x^2 \) (since it opens downward) or maybe a quadratic with reflection. Wait, the graph is a parabola opening downward, vertex at (2, 1) maybe? Wait, let's see. The standard form for a downward opening parabola is \( y = -a(x - h)^2 + k \). Let's assume vertex is at (2, 1), and it's a reflection over x-axis (since it opens down) and shifted. Wait, maybe the parent is \( y = -x^2 \), shifted right by 2 and up by 1. So transformation: reflection over x-axis, horizontal shift right by 2, vertical shift up by 1.

Step 2: Write the Equation

So \( f(x)=-(x - 2)^2 + 1 \).

Answer:

Part a:

• Transformation(s): Vertical shift down by 1 unit.
• \( f(x) = x^2 - 1 \)

Part b:

• Transformation(s): Horizontal shift left by 3 units.
• \( f(x) = (x + 3)^2 \)

Part c:

• Transformation(s): Horizontal shift right by 3 units, vertical shift down by 3 units.
• \( f(x) = (x - 3)^2 - 3 \)

Part d:

• Transformation(s): Reflection over x - axis, horizontal shift right by 2 units, vertical shift up by 1 unit.
• \( f(x) = -(x - 2)^2 + 1 \)