Quadratic Graphs

Quadratic Graphs

Quadratic graphs

A quadratic is a function of the form $y=a{x}^2+bx+c$ where $a$ is not zero They are a very common type of function in mathematics, so it is important to know their key features

What does a quadratic graph look like?

• The shape made by a quadratic graph is known as a parabola

• The parabola shape of a quadratic graph can either look like a “∪-shape” or a “∩-shape”

  • A quadratic with a positive coefficient of $x^2$ will be a ∪-shape

  • A quadratic with a negative coefficient of $x^2$ will be a ∩-shape

• A quadratic will always cross the $y$-axis

• A quadratic may cross the $x$-axis twice, once, or not at all

  • The points where the graph crosses the $x$-axis are called the roots

• If the quadratic is a ∪-shape, it has a minimum point (the bottom of the ∪)

• If the quadratic is a ∩-shape, it has a maximum point (the top of the ∩)

• Minimum and maximum points are both examples of turning points

How do I sketch a quadratic graph?

• We could create a table of values for the function and then plot it accurately

  • However we often only require a sketch to be drawn, showing just the key features

• The key features needed to be able to sketch a quadratic graph are

  • the overall shape

    • ∪-shape graphs occur when $a>0$ (positive quadratic)

    • ∩-shape graphs occur when $a<0$ (negative quadratic)

  • the $x$-intercept(s), these are also known as the roots (there may be none!)

    • roots are found by setting the quadratic function (or $y$) equal to zero

    • i.e. solve $a{x}^2+bx+c=0$

    • if there are no (real) solutions (i.e. no roots), the graph does not intersect the $x$-axis

      • the discriminant can be used to determine whether a quadratic function has 0, 1 or 2 roots

  • the $y$-intercept

    • this is found by setting $x=0$ in the quadratic function

    • so for $a{x}^2+bx+c$ the coordinates of the $y$-intercept will be $\left(0, c\right)$

  • the minimum or maximum point (turning point)

    • sometimes a rough idea of where this should lie is enough

    • sometimes the specific coordinates of the turning point will be needed

    • when required the coordinates of the turning point can be found by either completing the square or differentiation

      • in cases where the quadratic has just one root, the graph will touch (rather than cross) the $x$-axis and so this will be the turning point

Sketching Graphs by Completing the Square

Sketching graphs by completing the square

How does completing the square help me sketch graphs?

• Completing the square can quickly tell us the coordinates of the turning point on a quadratic graph

• This is based on the fact that a squared term (e.g.  ${\left(x+1\right)}^2$) cannot be negative

• STEP 1
  Complete the square - rewrite $a{x}^2+bx+c=0$ in the form $a{\left(x+p\right)}^2+q$

• STEP 2
  Deduce the $\mathit{x}$-coordinate of the turning point

• ${\left(x+p\right)}^2\ge 0$ for all values of $x$

  • Therefore it's minimum value is 0, and this occurs when $x=-p$

The $x$-coordinate is $-p$

• STEP 3
  Deduce the $\mathit{y}$-coordinate of the turning point

• $a{\left(x+p\right)}^2=0$

  • Therefore $y=q$

The $y$-coordinate is $q$

• STEP 4
  The turning point has coordinates $\left(-p, q\right)$ This can be considered when sketching the graph of the quadratic function

• Note that the turning point could be a maximum or minimum point - this will depend on the value of $a$

  • $a$ is the coefficient of the $x^2$ term

  • If $a$ is positive, the graph is $\mathbf{\cup }$- shaped and will have a minimum point

  • If $a$ is negative, the graph is $\mathbf{\cap }$- shaped and will have a maximum point

    

How do I use the graph of a quadratic function to find its range?

• The range of a quadratic function will be shown on its graph by the values $y$ takes

  • i.e.  the turning point from a quadratic graph will determine its range

• For the quadratic function $\mathrm{f}\left(x\right)$ whose graph has a minimum point $\left(x_{min}, y_{min}\right)$

  • the range of the $\mathrm{f}\left(x\right)$ will be $\mathrm{f}\ge y_{min}$

• For the quadratic function $\mathrm{f}\left(x\right)$ whose graph has a maximum point $\left(x_{max}, y_{max}\right)$

  • the range of $\mathrm{f}\left(x\right)$ will be $\mathrm{f}\le y_{max}$

• If there any restrictions on the domain of $\mathrm{f}\left(x\right)$ then they could affect the range of $\mathrm{f}\left(x\right)$