Question

If the following graph is of a function `f(x)`, how will the graph of (i) `f(x+3)`, (ii) `f(x)+3`, (iii) `-f(x)` and (iv) `1-f(x-3)` appear?

Answer 1

Please see below.

Explanation:

The graph appears to be of a parabola `y=f(x)=-x^2+2`
graph{-x^2+2 [-5, 5, -2.5, 2.5]}

(i) `y=f(x+3)=-(x+3)^2+2=-x^2-6x-9+2=-x^2-6x-7`
graph{-x^2-6x-7 [-5.896, 4.104, -1.9, 3.1]}

(ii) `y=f(x)+3=-x^2+2+3=-x^2+5`
graph{-x^2+5 [-10.775, 9.225, -3.76, 6.24]}

(iii) `y=-f(x)=-(-x^2+2)=x^2-2`
graph{x^2-2 [-10.73, 9.27, -2.2, 7.8]}

(iv) `y=1-f(x-3)=1-(-(x-3)^2+2)`
= `1+x^2-6x+9-2=x^2-6x+8`
graph{x^2-6x+8 [-7.77, 12.23, -1.72, 8.28]}