How to find the equation after stretching and shifting a quadratic graph

Q: What is the transformed equation of f(x)=x^2+3x-4 after a horizontal stretch by 2 and a shift down 3?

The transformed function is g(x)=f(x/2)-3, which simplifies to g(x)=x^2/4+3x/2-7.

Q: Why is f(x/2) used for a horizontal stretch by 2?

Replacing x with x/2 makes every x-coordinate twice as large, which creates a horizontal stretch by factor 2.

Question

6. $f(x)=x^2+3x-4$
Case A:
- horizontal stretch by a scale factor of 2
- translation of 3 units towards the negative y-axis
$f\left(\frac{x}{2}ight)-f\left(\frac{x}{2}ight)$
Case B: translation of 3 units toward the negative y-axis and horizontal stretch by a scale factor of 2.
$g(x)=\frac{x^2}{4}+\frac{3x}{2}-7$
$[x^2+3x-4]\;	ext{m}-3$
$x^2+3x-7$
$\left(\frac{x}{2}ight)^2+3\left(\frac{x}{2}ight)-7$
$\frac{x^2}{4}+\frac{3x}{2}-7$

Acemy AI · Accepted Answer

> **Expert intro**: When a quadratic is transformed, the safest method is to apply one change at a time and only expand at the end.

### Detailed walkthrough
Start with
$$
f(x)=x^2+3x-4.
$$

A horizontal stretch by factor 2 gives
$$
f\left(\frac{x}{2}\right)=\left(\frac{x}{2}\right)^2+3\left(\frac{x}{2}\right)-4.
$$

Then translating the graph down 3 units means subtract 3:
$$
g(x)=f\left(\frac{x}{2}\right)-3.
$$

Now simplify:
$$
g(x)=\frac{x^2}{4}+\frac{3x}{2}-4-3
=\frac{x^2}{4}+\frac{3x}{2}-7.
$$

So the transformed function is
$$
\boxed{g(x)=\frac{x^2}{4}+\frac{3x}{2}-7}.
$$

If you compare the two transformation orders in the prompt, the result is the same here because the vertical translation is just a constant shift.

### 💡 Pitfall guide
Do not replace $x$ with $2x$ for a stretch by factor 2. That reverses the effect. Also, avoid subtracting 3 inside the input; the downwards shift belongs outside the function.

### 🔄 Real-world variant
If the same graph were stretched horizontally by factor 3 instead, you would use $f(x/3)$. The final expression would become
$$
\left(\frac{x}{3}\right)^2+3\left(\frac{x}{3}\right)-4-3.
$$
The method stays the same; only the input scale changes.

### 🔍 Related terms
quadratic function, graph transformations, horizontal stretch

Question

How to find the equation after stretching and shifting a quadratic graph

Expert Verified Solution

Detailed walkthrough

FAQ

What is the transformed equation of f(x)=x^2+3x-4 after a horizontal stretch by 2 and a shift down 3?

Why is f(x/2) used for a horizontal stretch by 2?