What are Quadratic Functions? 03:19 minutes

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Transcript What are Quadratic Functions?

"You've seen the posters. You've seen the billboards. Now drink the drink. The question of the summer is about to be answered. The Quadratic Functions are proud to present a refreshing line of new drinks, Parabolica! Parabolica XTRA Strong is just amazing. It makes gravity work better!"

”That jump is a perfectly shaped parabola! What a gorgeous u-shape! You'll love the extra strong flavor. It’s simply refreshing. You wanna know Parabolica's secret ingredients? We're gonna tell ya. PARABOLAS! You may be wondering, is there a standard form like with linear functions? You betcha!”

A quadratic equation, written in standard form, is equal to y=ax²+bx+c for all 'a's not equal to zero.

”Wanna be a quadratic function, too? Then the 2nd degree is the way to be! Tame your inner beast, just look at that beautiful parabola! Also from quadratic functions: Parabolica Chill! Try it! Feel the thrill of Parabolica Chill! This drink will give you energy, but not quite so much."

Back to parabolas and quadratic equations. Let's investigate some characteristics of the equations and their corresponding graphs. The 'a' value, the coefficient of the squared variable, determines the width of the parabola. The greater the 'absolute' value of a, the thinner the parabola. This is the graph of the quadratic equation, with 'a' = 1: y = x². And this is a graph of a quadratic equation, with 'a' less than 1: y = (1/2)x². And here we have 'a' greater than 1: y = 6x². When we compare to the graph of y = x², it's almost as if were on a diet...

Ok, enough math, back to the energy drinks.
_”Something weighing you down? Try the new Parabolica Lite – you’ll feel light as a feather! You might even defy gravity!”

”Will you just take a look at that nicely-shaped parabola?! See how light he is after drinking Parabolica Lite? Wanna know another secret about Parabolas?”

The sign of the 'a' value affects a parabola's direction. When the 'a' value is negative, the parabola is shaped like a frown. But when the 'a' value is positive, we can turn that frown upside down!

”Now that we've told you all the secret ingredients for Parabolas" Oh dear! It seems they should have invested more money in getting just the right taste - rather than on special effects

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  • Decide if the given shape is a parabola.

    Tipps

    A parabola looks like a rainbow or a smile.

    A parabola doesn't have an angle.

    Three of the given examples are parabolas.

    Lösung

    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.

  • Determine which functions are quadratic functions.

    Tipps

    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:

    $y=\frac12x^2+x$.

    Lösung

    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 $b=c=0$.

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

    Tipps

    An example:

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

    Lösung

    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 $y=x^2$.

  • Explain how to find the corresponding quadratic function.

    Tipps

    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.

    Lösung

    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 $y=2x^2$.

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

    Tipps

    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.

    Lösung

    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.

    Tipps

    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 $y=ax^2$.

    Lösung

    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: $y=-\frac13\times(-3)^2=-\frac93=-3$. ✓