What are Quadratic Functions?

What are Quadratic Functions? exercise

• Decide if the given shape is a parabola.

  Hints

  A parabola looks like a rainbow or a smile.

  A parabola doesn't have an angle.

  Three of the given examples are parabolas.

  Solution

  A parabola is shaped like a rainbow.

  The standard form of an equation for a parabola is given by

  $y=ax^2+bx+c$.

  Depending on the sign of $a$ you can determine the direction of the parabola; the parabola can look like a rainbow itself or smile.

  There are three images showing a parabola:

  • The plane flying.
  • The dolphin jumping.
  • The swimmer jumping.

  The rollercoaster isn't a parabola at all.

• Describe a quadratic function and the role of $a$.

  Hints

  An example:

  $y=3x^2-2$ is an example of a quadratic function.

  Solution

  A quadratic function in standard form is given by

  $y=ax^2+bx+c$.

  Depending on $a$ you can decide the width of the parabola.

  For example, let's see when the parabola $y=ax^2$ is thinner than $y=x^2$, and when $y=ax^2$ is wider than $y=x^2$.

  The greater the absolute value of $a$, the thinner the parabola.

  So when $a>1$, the parabola $y=ax^2$ is thinner than $y=x^2$.

  And when $0<a<1$, the parabola $y=ax^2$ is wider than $y=x^2$.

• Identify the value of the coefficient $a$ of each parabola.

  Hints

  The greater the absolute value of $a$, the thinner the parabola.

  One of the parabolas is the graph of $y=x^2$.

  Each quadratic function has the form $y=ax^2$, with $b=c=0$.

  Check the $(x,y)$ pairs lying on the parabola.

  Solution

  The parabola to the right is the graph of $y=x^2$. So $a=1$.

  How can this be determined? Look at some $(x,y)$ points lying on the parabola: $(1,1)$, $(2,4)$, $(-1,1)$. All these points satisfy $y=x^2$.

  The smallest $a$ is given by $a=\frac14$. The parabola corresponding to $y=\frac14x^2$ could be determined by the points $(2,1)$ and $(4,4)$.

  Next we have $a=\frac12$, which can be determined via the points $(2,2)$ and $(1,0.5)$.

  Then we have $a=1$, which is pictured here to the right.

  Then we have the parabola which is the graph of $y=1.5x^2$; so $a=1.5$. This can be detected by the points $(1,1.5)$ or $(2,6)$.

  The last parabola corresponds to $y=3x^2$; $a=3$ and $(1,3)$ satisfies $y=3x^2$.

• Find the corresponding quadratic function.

  Hints

  Each quadratic function has the form $y=ax^2$.

  Keep the following in mind:

  • If $a>0$, then the parabola looks like a rainbow.
  • If $a<0$, then the parabola looks like a smile.

  Each pair $(x,y)$ on a parabola must satisfy $y=ax^2$.

  Solution

  Each of the parabolas above belongs to a quadratic function of the form $y=ax^2$.

  First we look at the sign of $a$: the yellow as well as the red parabola are opened downwards, or are "rainbow shaped". The blue and the green parabola are opened upwards, or are "smile shaped".

  Next we pick some points $(x,y)$ lying on each parabola:

  • For the blue parabola, we pick the point $(3,3)$, and check: $y=\frac13 3^2=\frac93=3$. ✓
  • For the green parabola, we pick the point $(1,2)$ and check: $y=2\times1^2=2\times 1=2$. ✓
  • For the red parabola, we pick the point $(1,-1)$ and check: $y=-1^2=-1$. ✓
  • For the yellow parabola, we pick the point $(-3,-3)$ and check: $y=-\frac13\times(-3)^2=-\frac93=-3$. ✓

• Determine which functions are quadratic functions.

  Hints

  A quadratic function means that the hightest power is $2$.

  For example, $y=mx+b$ is a linear function, where the highest power $1$.

  The following is an example of a quadratic function:

  $y=\frac12x^2+x$.

  Solution

  The standard form of a quadratic function is given by

  $y=ax^2+bx+c$.

  So the highest power is $2$ and $a\neq 0$. Otherwise we would have a linear function $y=mx+n$.

  With $a=1$ and $b=c=0$, the function $y=x^2$ is quadratic, as well as the functions:

  • $y=6x^2$ with $a=6$ and $b=c=0$.
  • $y=\frac12 x^2$ with $a=\frac12$ and $b=c=0$.

• Explain how to find the corresponding quadratic function.

  Hints

  The standard form of a quadratic function is given by

  $y=ax^2+bx+c$.

  The sign of $a$ determines if the corresponding parabola is opened upwards or downwards.

  The larger the absolute value of $a$, the thinner the parabola. Similarly, the smaller the absolute value of $a$, the wider the parabola.

  Solution

  Here you see a parabola, the graph of a quadratic function.

  How can you determine the corresponding equation?

  1. Keep the standard form in mind: $y=ax^2+bx+c$.
  2. The vertex of the parabola is the origin, so we have $b=c=0$ and thus that our equation is of the form $y=ax^2$.
  3. The parabola is opened upwards. Thus we know that $a$ must be positive.
  4. The parabola is thinner than the parabola corresponding to $y=x^2$ (i.e. the red parabola in the picture to the right). You can recognize this, for example, by looking at the point $(1,1)$ on $y=x^2$, and $(1,2)$ on the blue parabola in question. Thus $a$ must be greater than $1$.
  5. To determine $a$, plug a point on the parabola into $y=ax^2$. For example, the point $(2,8)$ gives us $8=a\times 2^2$. Dividing by $4$ results in $a=2$.

  And so the resulting quadratic function is $y=2x^2$.