Quadratic Functions (DP IB Analysis & Approaches (AA): HL): Revision Note

Quadratic functions & graphs

What are the key features of quadratic graphs?

• A quadratic graph can be written in 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 concave up $\cup$

    • The graph has a minimum point

  • If $a$ is negative the shape is concave down $\cap$

    • The graph has a maximum point

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

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

  • These can be found by

    • Factorising

    • Quadratic formula

    • Using your GDC

  • These are also called the $x$-intercepts

  • A quadratic graph can have 0, 1 or 2 $x$-intercepts

    • This is determined by the value of the discriminant

• 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 their midpoint

    • E.g. If there are roots at $\left(2,0\right)$ and $\left(4,0\right)$ then the axis of symmetry is at $\left(3,0\right)$

• The vertex lies on the axis of symmetry

  • It can be found by completing the square

  • The $x$-coordinate of the vertex is $x=-\frac{b}{2a}$

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

  • If $a$ is positive then the vertex is the minimum point

  • If $a$ is negative then the vertex is the maximum point

What are the equations of a quadratic function?

• $f\left(x\right)=a{x}^2+bx+c$

  • This is the general form

  • It clearly shows the $y$-intercept $(0,c)$

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

    • This is given in the formula booklet

• $f\left(x\right)=a(x-p)(x-q)$

  • This is the factorised form

  • It clearly shows the roots $\left(p,0\right)$ and $(q,0)$

  • The axis of symmetry is $x=\frac{p+q}{2}$

• $f\left(x\right)=a{(x-h)}^2+k$

  • This is the vertex form

  • It clearly shows the vertex $(h,k)$

  • The axis of symmetry is therefore $x=h$

  • It clearly shows how the function can be transformed from the graph $y=x^2$

    • Vertical stretch by scale factor ­$a$

    • Translation by vector $\left(\begin{array}{c}h\\ k\end{array}\right)$

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 or by completing the square

  • e.g. $2{\left(-0.75\right)}^2+3\left(-0.75\right)-2=-3.125$ or $2x^2+3x-2=2{\left(x+\frac{3}{4}\right)}^2-\frac{25}{8}$

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

• Find the roots by setting the expression equal to zero and solving

  • 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

How do I find an equation of a quadratic?

• If you have the roots $x=p$ and $x=q$,

  • Write in factorised form$y=a(x-p)(x-q)$

  • You will need a third point on the curve to substitute in to find the value of $a$

• If you have the vertex $(h,k)$,

  • Write in vertex form $y=a{(x-h)}^2+k$

  • You will need a second point on the curve to find the value of $a$

• If you have three random points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$,

  • Write in the general form $y=a{x}^2+bx+c$

  • Substitute the three points into the equation, one at a time

  • Form and solve a system of three linear equations to find the values of $a$, $b$ and $c$