Question: User: (a) Draw the graph of y = f(x + 1) + 2 onto the grid above. (2)
Point A(-2, 2) is on the graph y = f(x)
When the graph of y = f(x) is transformed to the graph with equation y = f(-x) the point A is mapped to point B.
(b) Write down the coordinates of point B.
graph: y = f(x + 1) + 2, position_hint {top-left, top-right, center, bottom-left, bottom-right}
graph: point A(-2, 2) on y = f(x), position_hint {top-left, top-right, center, bottom-left, bottom-right}
graph: y = f(-x), point A mapped to point B, position_hint {top-left, top-right, center, bottom-left, bottom-right}
1. **State the problem:**
We have a function $y = f(x)$ with a point $A(-2, 2)$ on its graph.
We want to:
- (a) Draw the graph of the transformed function $$y = f(x + 1) + 2$$.
- (b) Find the coordinates of point $B$ when the graph is transformed to $$y = f(-x)$$, where $A$ maps to $B$.
2. **Understanding transformations:**
- The transformation $$y = f(x + 1) + 2$$ shifts the graph of $f(x)$ **left by 1 unit** (because of $x+1$ inside the function) and **up by 2 units** (because of $+2$ outside the function).
- The transformation $$y = f(-x)$$ reflects the graph of $f(x)$ about the **y-axis**.
3. **Drawing the graph of $$y = f(x + 1) + 2$$:**
- Every point $(x, y)$ on $y = f(x)$ moves to $(x - 1, y + 2)$ on $y = f(x + 1) + 2$.
4. **Mapping point A for part (a):**
- Original point $A$ is $(-2, 2)$.
- After transformation, new point $A'$ is:
$$ig((-2) - 1, 2 + 2ig) = (-3, 4)$$.
5. **Finding point B for part (b):**
- The transformation $$y = f(-x)$$ reflects points about the y-axis.
- For point $A(-2, 2)$, reflection about y-axis changes $x$ to $-x$:
$$B = (2, 2)$$.
**Final answers:**
- (a) The graph of $$y = f(x + 1) + 2$$ is the graph of $f(x)$ shifted left by 1 and up by 2.
- (b) The coordinates of point $B$ are $$\boxed{(2, 2)}$$.