Question

How can you use transformations to graph this function?
es002-1.jpg

Explain your steps.

Answer 1

I can't see the image file you attached. Please either paste the equation here or upload the image again.

While you do that, here is a short, general recipe you can use to graph any function by transformations (plus a worked example).

General steps
1. Identify the parent (base) function f(x) — e.g. x^2, √x, |x|, sin x, 1/x, etc.
2. Rewrite the function in a standard transformed form if needed, e.g.
y = a · f(b(x − h)) + k
where
- h is a horizontal shift,
- b affects horizontal stretch/compression and/or reflection in y-axis,
- a affects vertical stretch/compression and/or reflection in x-axis,
- k is a vertical shift.
3. Apply transformations in the correct conceptual order:
- Inside-x operations first (horizontal stretch/compression and horizontal shift, and any reflection across y-axis).
- Then apply vertical stretch/compression and reflection (multiply by a).
- Finally apply vertical shift (add k).
(This order reflects that inside changes act on x before f, outer changes act on outputs.)
4. Track a few key points and important features (vertex, intercepts, asymptotes, domain endpoints) from the parent graph through each transformation.
5. Sketch the result and check by plotting the transformed key points.

Worked example
Graph y = −2(x + 1)^2 + 3 using f(x) = x^2.

1. Parent: y = x^2 (a parabola with vertex (0,0)).
2. Rewriting shows h = −1 (inside is x − (−1)), b = 1, a = −2, k = 3.
3. Apply transformations in order:
- Horizontal shift: move the parabola left 1 unit → vertex at (−1, 0).
- Vertical stretch & reflection: multiply y-values by −2. The parabola becomes narrower (stretched by factor 2) and flips upside down. Vertex remains (−1, 0).
- Vertical shift: move up 3 units → vertex at (−1, 3).
4. Map a couple of points: from parent (1,1) → after shift left 1 becomes (0,1) → after vertical ×(−2) becomes (0, −2) → after +3 becomes (0,1). So (0,1) is on the final graph. Similarly map (2,4): (1,4) → (1, −8) → (1, −5).
5. Sketch using the transformed vertex, axis of symmetry x = −1, and the mapped points.

If you post the exact function from your image I’ll give step-by-step transformations for that specific problem.