Quadratic Functions & Graphs (DP IB Applications & Interpretation (AI): HL): Revision Note

Quadratic functions & graphs

What does a quadratic graph look like?

• A quadratic graph is of the form $y=a{x}^2+bx+c$ where $a\ne 0$.

• The value of $a$ affects the shape of the curve

  • If $a$ is positive the shape is $\cup$ and it has a minimum point

  • If $a$ is negative the shape is $\cap$ and it has a maximum point

What are the key features of a quadratic graph?

• The y-intercept is at the point $\left(0,c\right)$

• The zeros or roots are the solutions to $a{x}^2+bx+c=0$

  • These can be found using your GDC or the quadratic formula

  • These are also called the x-intercepts

  • There can be 0, 1 or 2 x-intercepts

• There is an axis of symmetry at $x=-\frac{b}{2a}$

  • This is given in your formula booklet

  • If there are two x-intercepts then the axis of symmetry goes through the midpoint of them

• The vertex lies on the axis of symmetry

  • The x-coordinate is $-\frac{b}{2a}$

  • The y-coordinate can be found using the GDC or by calculating y when $x=-\frac{b}{2a}$

How do I sketch a quadratic graph?

• Determine its shape by looking at the value of $a$

  • e.g. $y=2x^2+3x-2$ looks like $\cup$

• Find the axis of symmetry using the formula

  • e.g. $x=-\frac{3}{2\left(2\right)}=-0.75$

• Find the vertex by substituting the x-coordinate into the equation

  • e.g. $2{\left(-0.75\right)}^2+3\left(-0.75\right)-2=-3.125$

  • vertex is at $\left(-0.75,-3.125\right)$

• Find the roots using your GDC

  • e.g. $2x^2+3x-2=0\Rightarrow x=0.5\mathrm{or}x=-2$

  • roots are $\left(0.5,0\right)$ and $\left(-2,0\right)$

• Label the y-intercept

  • e.g. $\left(0,-2\right)$

• Draw the graph going through all the labelled points