Question

Given the function f(x) = x3 + x2 − 2x + 1, what is the resulting function when f(x) is shifted to the left 1 unit?
A f(x − 1) = x3 − 2x2 − x + 3
B f(x + 1) = x3 + 4x2 + 3x + 1
C f(x) − 1 = x3 + x2 − 2x
D f(x) + 1 = x3 + x2 − 2x + 2

Answer 1

The answer is B.

Explanation:

If c is a positive real number, then the graph of
f(x – c) is the graph of y = f(x) shifted to the right
c units.
Horizontal Shifts
If c is a positive real
number, then the
graph of f(x + c) is
the graph of y = f(x)
shifted to the left

Answer 2

Answer:

Option B is correct

Step-by-step explanation:

Given: $f(x)=x^3+x^2-2x+1$

If a graph $f(x)$ is shifted a units to the left then it becomes $f(x+a)$

Here, the graph is shifted 1 unit to the left . On taking a=1, we get graph $f(x+1)$

Using formula $\left ( a+b \right )^{3}=a^3+b^3+3a^2b+3ab^2$ , we get

$\left ( x+1 \right )^{3}=x^3+1+3x^2+3x$

Using formula $(a+b)^{2}=a^2+b^2+2ab$ , we get

$\left ( x+1 \right )^{2}=x^2+1+2x$

Using distributive property over multiplication i.e $a\left ( b+c \right )=ab+ac$ , we get

$2\left ( x+1 \right )=2x+2$

Therefore,

$f(x)=x^3+x^2-2x+1\\\Rightarrow f\left ( x+1 \right )=\left ( x+1 \right )^{3}+\left ( x+1 \right )^{2}-2\left ( x+1 \right )+1\\=x^3+1+3x^2+3x+x^2+2x+1-2x-2+1\\=x^3+4x^2+3x+1$