Simple Quadratic Equations

In $y=a{x}^2+bx+c$, the sign of a decides which way the curve opens.

When a is positive, $a{x}^2$ is positive for every non-zero value of x, so for large positive x and for large negative x both arms of the curve rise. The parabola opens upward, making a U shape with a minimum turning point.

The curve is still symmetrical: it has a vertical line of symmetry through its turning point.

The size of a (how far it is from zero) controls how narrow or wide the curve is, not which way it opens.

When a is negative, $a{x}^2$ is negative for every non-zero value of x, so both arms of the curve fall. The parabola opens downward, making an n shape with a maximum turning point.

The curve is still symmetrical about a vertical line through its turning point - a negative a flips the curve, it does not distort it.

How wide or narrow the curve is depends on the size of a (its absolute value), so without knowing the actual value of a you cannot say whether it is wider or narrower than $y=x^2$.

$a=4$ is positive, so the parabola still opens upward.

Every y-value is 4 times as large as on $y=x^2$: at $x=1$ the curve reaches $y=4$ instead of $y=1$, and at $x=2$ it reaches $y=16$ instead of $y=4$.

Rising 4 times as fast makes the curve steeper, so it is narrower than $y=x^2$. In general $|a|>1$ gives a narrower curve and $|a|<1$ gives a wider one.

The coefficient of $x^2$ stretches the curve vertically; it does not slide the curve up or down. That is the job of the constant term.

$a=0.5$ is positive, so the parabola still opens upward.

Every y-value is half as large as on $y=x^2$: at $x=2$ the curve reaches $y=2$ instead of $y=4$, and at $x=4$ it reaches $y=8$ instead of $y=16$.

Rising half as fast makes the curve flatter, so it is wider than $y=x^2$. In general $|a|<1$ gives a wider curve and $|a|>1$ gives a narrower one.

The coefficient of $x^2$ stretches or squashes the curve vertically; it does not slide it up or down.

$a=-2$ is negative, so the parabola is flipped over and opens downward.

The size of a is $|-2|=2$, so every point is twice as far from the x-axis as on $y=x^2$: at $x=1$ the curve reaches $y=-2$ instead of $y=1$, and at $x=2$ it reaches $y=-8$ instead of $y=4$.

Because $|a|>1$ the curve falls twice as steeply, so it is narrower than $y=x^2$.

So the sign of a flips the curve and the size of a stretches it. The coefficient does not move the curve up or down.

$a=-0.5$ is negative, so the parabola is flipped over and opens downward.

The size of a is $|-0.5|=0.5$, which is less than 1, so every point is only half as far from the x-axis as on $y=x^2$: at $x=2$ the curve reaches $y=-2$ instead of $y=4$, and at $x=4$ it reaches $y=-8$ instead of $y=16$.

Falling only half as fast makes the curve flatter, so it is wider than $y=x^2$.

So the sign of a flips the curve and the size of a decides the width: $|a|<1$ gives a wider curve, $|a|>1$ a narrower one.