Math / Algebra

QuestionWrite the equation $g(x)$ in vertex form of a quadratic function for the transformations given the function $f(x)=x^{2}$ . SEE EXAMPLE 5
30. Let $g(x)$ be the function whose graph is a translation 4 units left and 1 unit up of the graph of $f(x)$ .
31. Let $g(x)$ be the function whose graph is a reflection in the $x$ -axis and translated 3 units right of the graph of $f(x)$ .

Studdy Solution

STEP 1

1. The function $f(x) = x^2$ is a standard quadratic function.
2. Transformations include translations (shifts) and reflections.
3. Vertex form of a quadratic function is $g(x) = a(x-h)^2 + k$ , where $(h, k)$ is the vertex.
4. Translation 4 units left implies $h = -4$ .
5. Translation 1 unit up implies $k = 1$ .
6. Reflection in the $x$ -axis implies $a = -1$ .
7. Translation 3 units right implies $h = 3$ .

STEP 2

1. Translate $f(x)$ 4 units left and 1 unit up to find $g(x)$ .
2. Reflect $f(x)$ in the $x$ -axis and translate it 3 units right to find $g(x)$ .

STEP 3

Translate $f(x)$ 4 units left. This transformation affects the $x$ -variable by replacing $x$ with $(x+4)$ .
$f(x+4) = (x+4)^2$

STEP 4

Translate the resulting function $1$ unit up by adding $1$ to the entire function.
$g(x) = (x+4)^2 + 1$

STEP 5

The vertex form of the translated function is:
$g(x) = (x+4)^2 + 1$

STEP 6

Reflect $f(x)$ in the $x$ -axis. This transformation affects the $y$ -variable by multiplying it by $-1$ .
$-f(x) = -x^2$

STEP 7

Translate the resulting function $3$ units right. This transformation affects the $x$ -variable by replacing $x$ with $(x-3)$ .
$g(x) = -(x-3)^2$

STEP 8

The vertex form of the reflected and translated function is:
$g(x) = -(x-3)^2$

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Studdy Solution

STEP 1

STEP 2

1. Translate $f(x)$ 4 units left and 1 unit up to find $g(x)$ .
2. Reflect $f(x)$ in the $x$ -axis and translate it 3 units right to find $g(x)$ .

STEP 3

Translate $f(x)$ 4 units left. This transformation affects the $x$ -variable by replacing $x$ with $(x+4)$ .
$f(x+4) = (x+4)^2$

STEP 4

Translate the resulting function $1$ unit up by adding $1$ to the entire function.
$g(x) = (x+4)^2 + 1$

STEP 5

The vertex form of the translated function is:
$g(x) = (x+4)^2 + 1$

STEP 6

Reflect $f(x)$ in the $x$ -axis. This transformation affects the $y$ -variable by multiplying it by $-1$ .
$-f(x) = -x^2$

STEP 7

Translate the resulting function $3$ units right. This transformation affects the $x$ -variable by replacing $x$ with $(x-3)$ .
$g(x) = -(x-3)^2$

STEP 8

The vertex form of the reflected and translated function is:
$g(x) = -(x-3)^2$

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Studdy Solution

STEP 1

STEP 2

1. Translate f(x)f(x)f(x) 4 units left and 1 unit up to find g(x)g(x)g(x).2. Reflect f(x)f(x)f(x) in the xxx-axis and translate it 3 units right to find g(x)g(x)g(x).

STEP 3

Translate f(x)f(x)f(x) 4 units left. This transformation affects the xxx-variable by replacing xxx with (x+4)(x+4)(x+4).f(x+4)=(x+4)2 f(x+4) = (x+4)^2 f(x+4)=(x+4)2

STEP 4

Translate the resulting function 111 unit up by adding 111 to the entire function.g(x)=(x+4)2+1 g(x) = (x+4)^2 + 1 g(x)=(x+4)2+1

STEP 5

The vertex form of the translated function is:g(x)=(x+4)2+1 g(x) = (x+4)^2 + 1 g(x)=(x+4)2+1

STEP 6

Reflect f(x)f(x)f(x) in the xxx-axis. This transformation affects the yyy-variable by multiplying it by −1-1−1.−f(x)=−x2 -f(x) = -x^2 −f(x)=−x2

STEP 7

Translate the resulting function 333 units right. This transformation affects the xxx-variable by replacing xxx with (x−3)(x-3)(x−3).g(x)=−(x−3)2 g(x) = -(x-3)^2 g(x)=−(x−3)2

STEP 8

The vertex form of the reflected and translated function is:g(x)=−(x−3)2 g(x) = -(x-3)^2 g(x)=−(x−3)2

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1. Translate $f(x)$ 4 units left and 1 unit up to find $g(x)$ .
2. Reflect $f(x)$ in the $x$ -axis and translate it 3 units right to find $g(x)$ .

Translate $f(x)$ 4 units left. This transformation affects the $x$ -variable by replacing $x$ with $(x+4)$ .
$f(x+4) = (x+4)^2$

Translate the resulting function $1$ unit up by adding $1$ to the entire function.
$g(x) = (x+4)^2 + 1$

The vertex form of the translated function is:
$g(x) = (x+4)^2 + 1$

Reflect $f(x)$ in the $x$ -axis. This transformation affects the $y$ -variable by multiplying it by $-1$ .
$-f(x) = -x^2$

Translate the resulting function $3$ units right. This transformation affects the $x$ -variable by replacing $x$ with $(x-3)$ .
$g(x) = -(x-3)^2$

The vertex form of the reflected and translated function is:
$g(x) = -(x-3)^2$