Year 10 Mathematics Core · Cambridge Ch 7, §7D · Mr Wong
Name
Class
Date
Learning intentions.
Use completing the square to write any quadratic in turning point form $y=(x-h)^2+k$
Read the turning point $(h,k)$ and axis of symmetry; find $x$- and $y$-intercepts
Sketch the parabola with key features labelled
Key results
To complete the square on $x^2+bx$, add and subtract $\left(\tfrac{b}{2}\right)^2$.
In $y=(x-h)^2+k$: turning point $(h,k)$, axis of symmetry $x=h$.
$y$-intercept: put $x=0$. $x$-intercepts: put $y=0$ and take $\pm\sqrt{\ }$ of both sides.
Warm-up — completing the square
1. Fill in the gaps. $y=x^2+2x-5 = x^2+2x+\underline{\hspace{8mm}}-\underline{\hspace{8mm}}-5 = (x+\underline{\hspace{6mm}})^2-\underline{\hspace{8mm}}$ so TP $=(\underline{\hspace{8mm}},\underline{\hspace{8mm}})$.
2. Solve for $x$, giving exact answers.
a $x^2=9$
b $x^2=3$
c $(x-1)^2=16$
d $(x+4)^2=2$
Part A — Turning point form
3. Write each in turning point form $y=(x-h)^2+k$, then state the turning point.
a $y=x^2-6x+10$
b $y=x^2+4x+1$
c $y=x^2-8x+20$
d $y=x^2+10x+18$
4. For $y=-3(x-2)^2+12$ state: a the turning point (max or min), b the $y$-intercept, c the $x$-intercepts.
a
b
c
Part B — Sketching
5. For $y=x^2+6x+15$: complete the square, then state the turning point, the $y$-intercept and the number of $x$-intercepts. Sketch on the grid.
a TP form
b Turning point
c $y$-intercept
d Number of $x$-intercepts
Parabolas 7D — continued
Part B (sketching) · Part C (intercepts in exact form) · Challenge
Name
Part B — Sketching (continued)
6. For $y=x^2-4x+2$: complete the square, find the turning point, the $y$-intercept and the $x$-intercepts in exact form, then sketch on the grid.
a TP form
b Turning point
c $y$-intercept
d $x$-intercepts (exact)
Part C — Intercepts in exact form
7. Complete the square and give the $x$-intercepts in exact (surd) form. State "none" if there are no $x$-intercepts.
a $y=x^2-2x-4$
b $y=x^2+2x+5$
c $y=x^2-6x+4$
d $y=x^2-3x-1$
Challenge
8. The parabola $y=x^2+px+q$ has its turning point at $(3,-4)$.