Graphing Inverse Functions - Starting Points Maths - Free Printable

Since all options (ExplainAnswer, TeachTopic, GeneratePractice) are false, I will only provide the final answer — but per instructions, I must first reason through the problem step by step.

The task is: *For each of the following, sketch the graph of the inverse function.*

There are six graphs labeled ① to ⑥. To sketch the inverse of a function from its graph, we reflect the graph over the line $ y = x $. That means:
- Every point $(a, b)$ on the original graph becomes $(b, a)$ on the inverse graph.
- The shape flips across the diagonal line $y = x$.

So for each graph, we need to find key points, swap their coordinates, and sketch the reflected curve.

Let’s go one by one:

---

①
Original points given: $(-3, 0)$, $(0, 6)$
→ Inverse points: $(0, -3)$, $(6, 0)$
The original is a straight line (linear), so inverse is also a straight line through those two points. Slope of original: $\frac{6 - 0}{0 - (-3)} = \frac{6}{3} = 2$.
Inverse slope should be reciprocal: $ \frac{1}{2} $. Check: from $(0, -3)$ to $(6, 0)$: slope = $\frac{0 - (-3)}{6 - 0} = \frac{3}{6} = \frac{1}{2}$. ✔
So inverse is line through $(0, -3)$ and $(6, 0)$.

②
Points: $(0, 8)$, $(4, 0)$
→ Inverse: $(8, 0)$, $(0, 4)$
Original slope: $\frac{0 - 8}{4 - 0} = -2$ → inverse slope = $-\frac{1}{2}$
Check: from $(8, 0)$ to $(0, 4)$: slope = $\frac{4 - 0}{0 - 8} = \frac{4}{-8} = -\frac{1}{2}$. ✔
Line through $(8, 0)$ and $(0, 4)$.

③
Points: $(0, 1)$, $(1, 5)$
→ Inverse: $(1, 0)$, $(5, 1)$
Original looks like exponential or increasing curve; inverse will be logarithmic-like (increasing, concave down). Passes through $(1, 0)$ and $(5, 1)$.

④
Points: $(0, 1)$, $(1, 0)$, $(4, 1)$
Wait — $(0,1)$ and $(1,0)$ suggest symmetry across $y=x$, and $(4,1)$ → inverse point is $(1,4)$. So inverse passes through $(1,0)$, $(0,1)$, $(1,4)$? But that would mean two points with same x=1 → not a function! Hmm — check: original graph likely passes through $(0,1)$, $(1,0)$, and $(4,1)$ — but if it's a function, it can’t have two y-values for same x, and here x=0→y=1, x=1→y=0, x=4→y=1 — that’s fine. Inverse would have:
$(1,0)$, $(0,1)$, $(1,4)$ — now x=1 maps to both y=0 and y=4 → not a function. That suggests original is not one-to-one, so inverse is not a function — but the problem asks to sketch the inverse relation anyway. So we still reflect all points.

But perhaps the point labeled $(4,1)$ is misread — looking back: in graph ④, point labeled $(4,1)$ is on the curve, and $(1,0)$ and $(0,1)$ are also marked. So yes, original is not one-to-one (fails horizontal line test), so inverse fails vertical line test — still, reflect all points.

So inverse has points: $(1,0), (0,1), (1,4)$ — meaning a curve going through those, likely with a vertical tangent or cusp at x=1.

But since this is a sketching task, we just reflect the whole graph over $y=x$.

⑤
Points: $(3,0)$, $(7,2)$
→ Inverse: $(0,3)$, $(2,7)$
Original is increasing curve (looks like square root or similar). Inverse will be increasing, steeper — passes through $(0,3)$ and $(2,7)$.

⑥
Points: $(-2, -8)$, $(2, 8)$
→ Inverse: $(-8, -2)$, $(8, 2)$
Original looks like cubic (odd function, passes through origin). Inverse of odd function is odd, and cubic inverse is also increasing, passing through those swapped points.

But the question says: *Sketch the graph of the inverse function* — and since this is a text-based response, and the user expects only the final answer (and options are false), I suspect the expected output is not sketches (which can’t be drawn here), but perhaps identification — but no, the instruction says: “Solve the problem accurately” and “Provide only the final correct answer”.

Wait — re-read: The user uploaded an image with 6 graphs, and the task is to sketch the inverse for each. Since we cannot draw here, and all flags are false, the only reasonable interpretation is that the system expects us to recognize that the correct approach is reflection over $y = x$, and perhaps the “final answer” is a description — but instructions say: *Only then provide the final answer*, and “final answer” should be just the answer.

However, in many such problems (especially from websites like startingpointsmaths.com), the expected output for a sketching task in a text format is often a list of the inverse point coordinates or identification.

But the problem says: “For each of the following, sketch the graph…” — and since we can’t sketch, and no multiple choice is given, the only logical conclusion is that the assistant is supposed to realize that without a way to output images, and with all options false, the appropriate action is to return nothing except the final answer — but that contradicts “must provide final answer”.

Let me double-check: maybe the actual expected answer is simply stating that the inverse graphs are reflections over $y = x$, and for grading purposes, the correct sketches correspond to swapping x and y of each labeled point.

Given the constraints, and that the user wants only the final answer after reasoning, and this is a homework problem where the student must sketch, but we’re an AI responding textually, the most appropriate “final answer” is to list the inverse point sets for each, as that fully determines the sketch.

So:

① Inverse points: $(0, -3), (6, 0)$
② Inverse points: $(8, 0), (0, 4)$
③ Inverse points: $(1, 0), (5, 1)$
④ Inverse points: $(1, 0), (0, 1), (1, 4)$
⑤ Inverse points: $(0, 3), (2, 7)$
⑥ Inverse points: $(-8, -2), (8, 2)$

But note: for ④, $(1,0)$ and $(1,4)$ conflict — however, the original graph likely has a point at $(4,1)$, not $(1,4)$ — wait, label says $(4,1)$ on graph ④, so yes, inverse has $(1,4)$. And $(1,0)$ comes from $(0,1)$? No: original $(0,1)$ → inverse $(1,0)$; original $(1,0)$ → inverse $(0,1)$; original $(4,1)$ → inverse $(1,4)$. So three points: $(1,0), (0,1), (1,4)$. That’s correct for the relation.

Since the problem likely expects the sketches, and we can’t draw, but the instruction says “ONLY THEN provide the final answer”, and no format is specified, I will give the answer as the set of inverse graphs described by swapped coordinates.

However, looking at standard practice for such problems on that website, sometimes the answer is just “reflect each graph over the line $y = x$”.

But the user said: “Solve the problem accurately.” And “final answer” should be concise.

Given all, the most precise final answer is:

The inverse graphs are obtained by reflecting each given graph across the line $y = x$, which swaps the coordinates of all points.

But that’s not a boxed answer.

Wait — perhaps the question is multiple choice in the original image, but user didn’t include choices. Since we have no choices, and options are false, the only safe move is: the problem asks to sketch, which we can’t do, so the correct response is to state the method — but instructions forbid explanation if ExplainAnswer is false.

Re-reading the top:
"→ If an option is false, completely omit that section."
All options are false, so we must only output:

Final Answer:
[answer]

With no explanation.

Given that, and to comply, I will provide the inverse point mappings as the answer — it’s the concrete result.

Final Answer:
① through (0, −3) and (6, 0); ② through (8, 0) and (0, 4); ③ through (1, 0) and (5, 1); ④ through (1, 0), (0, 1), and (1, 4); ⑤ through (0, 3) and (2, 7); ⑥ through (−8, −2) and (8, 2)