Graphing a circle and a parabola, then checking which one is a function

Expert intro: This question mixes graphing, the vertical line test, and a little geometric reasoning. The student’s comment about distance is not something you guess; it has to be visible from the shape of the graphs.

Detailed walkthrough

(a) Sketch the two relations

1. $x^2+y^2=9$

   • This is a circle centered at the origin.
   • Radius: $\sqrt{9}=3$.
   • It crosses the axes at $(\pm 3,0)$ and $(0,\pm 3)$.

2. $y=x^2+4$

   • This is a parabola opening upward.
   • Vertex: $(0,4)$.
   • Axis of symmetry: the $y$-axis.

(b) Which relation is a function?

The relation $y=x^2+4$ is a function because each $x$ value gives exactly one $y$ value.

The circle $x^2+y^2=9$ is not a function of $x$ because many vertical lines hit it twice. For example, when $x=0$, the equation gives $y=\pm 3$.

(c) Why is the minimum distance 1 unit?

Look at the top of the circle and the bottom of the parabola:

• Highest point on the circle: $(0,3)$
• Lowest point on the parabola: $(0,4)$

These points lie on the same vertical line, so the distance between them is

$4-3=1.$

Since the parabola stays above the circle near that region, this is the minimum distance.

(d) Another parabola with the same axis of symmetry and concave down

A parabola with the same line of symmetry as $y=x^2+4$ must have axis $x=0$. To make it concave down, the coefficient of $x^2$ must be negative. One valid example is

$y=-x^2+4.$

💡 Pitfall guide

A common mistake is to say the circle is a function because it has a neat formula. The vertical line test is the real check. Another trap is thinking the minimum distance must come from the nearest-looking edges; here, the vertical gap between $(0,3)$ and $(0,4)$ is the one that matters.

🔄 Real-world variant

If the parabola were shifted, say $y=x^2+5$, then the minimum distance to the circle would become 2 units instead of 1, because the lowest point of the parabola would be at $(0,5)$. If the parabola opened downward but kept the same axis, any equation of the form $y=-ax^2+k$ with $a>0$ would work.

🔍 Related terms

vertical line test, axis of symmetry, minimum distance