How to tell if one quadratic is a horizontal shift of another

Key takeaway: When two quadratics differ only in the sign of the linear term, the fastest check is to complete the square and compare the vertex positions.

We have

$$f(x)=2x^2-3x,\qquad g(x)=2x^2+3x.$$

Complete the square for both.

For $f$:

$$f(x)=2\left(x^2-\frac32x\right) =2\left[\left(x-\frac34\right)^2-\frac{9}{16}\right] =2\left(x-\frac34\right)^2-\frac98.$$

For $g$:

$$g(x)=2\left(x^2+\frac32x\right) =2\left[\left(x+\frac34\right)^2-\frac{9}{16}\right] =2\left(x+\frac34\right)^2-\frac98.$$

So the graphs have the same shape and the same $y$-value at the vertex, but the vertex of $g$ is shifted left by $\frac32$ units relative to $f$.

That means

$$g(x)=f\left(x+\frac32\right).$$

So the transformation is a horizontal shift left by $\frac32$ units.

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Pitfalls the pros know 👇 It is easy to mix up left and right when the formula is written as $f(x+a)$. Inside the function, a plus sign means shift left, not right. Another mistake is trying to compare coefficients directly without checking the vertex form, which makes the translation much clearer.

What if the problem changes? If the linear term in $g(x)$ were $-3x$ instead of $+3x$, then $g$ would be exactly the same as $f$. If the coefficient in front of $x^2$ changed too, then the graph would not be a pure translation anymore; you would also have a vertical stretch or compression.

Tags: horizontal shift, vertex form, complete the square