Find a parabola and symmetry line from a graphing-distance condition

Key takeaway: Once the axis of symmetry is fixed, the main job is to pick the right shape and opening. The equation does not have to be unique unless the question adds extra conditions.

(e)(i) One possible parabola

The parabola $y=x^2+4$ has axis of symmetry $x=0$. A parabola in the form $y=ax^2+c$ has the same axis of symmetry, and it is concave down when $a<0$.

One valid answer is

$y=-x^2+4.$

It shares the same line of symmetry and opens downward.

(e)(ii) Line of symmetry

For any parabola of the form

$y=ax^2+c,$

the line of symmetry is

$x=0.$

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Pitfalls the pros know 👇 Do not write a sideways parabola such as $x=y^2+4$ if the question asks for the same symmetry line as $y=x^2+4$. Also, concave down means the graph opens downward, so the coefficient of $x^2$ must be negative.

What if the problem changes? If the problem had asked for a different concave-down parabola with the same symmetry line, $y=-2x^2+4$ or $y=-\tfrac12 x^2+4$ would also work. Any $y=ax^2+c$ with $a<0$ keeps the axis at $x=0$.

Tags: axis of symmetry, concave down, parabola equation