Year 10 Mathematics Core · Cambridge Ch 7, §7G · Mr Wong
Name
Class
Date
Learning intentions.
Understand a line can meet a parabola at 0, 1 or 2 points
Find the points of intersection using substitution
Use the discriminant to find the number of intersection points
Key results
Substitution: substitute one equation into the other → $ax^2+bx+c=0$ → solve → substitute back for $y$.
Discriminant $b^2-4ac$: $<0$ → no points (miss), $=0$ → one point (tangent), $>0$ → two points (secant).
Give each answer as a coordinate pair $(x,y)$.
Warm-up
1. Find where each vertical line meets the parabola (give the point).
a $x=2$ on $y=2x^2+5x-6$
b $x=-1$ on $y=x^2+3x-1$
2. Rearrange into the form $ax^2+bx+c=0$ (with $a>0$).
a $x^2+5x=2x-6$
b $x^2-3x+4=2x+1$
Part A — Solving by substitution
3. Find the points of intersection (factorise where possible). State "none" if there are none.
a $y=x^2$, $y=3x$
b $y=x^2-2$, $y=x$
c $y=x^2+1$, $y=2x$
4. Solve $y=x^2$ and $y=2x-1$ simultaneously. How many points of intersection are there, and what is the line called?
Part B — Showing working & sketching
5. Find where $y=x^2-x-2$ meets $y=x+1$, then mark both points on the grid and sketch.
a Quadratic formed
b $x$-values
c Points of intersection
Parabolas 7G — continued
Part C (quadratic formula) · Part D (discriminant) · Challenge
Name
Part C — Using the quadratic formula
6. Solve $y=x^2+2x-1$ and $y=x+3$ simultaneously, rounding $x$ and $y$ to 2 d.p.
a Quadratic formed
b Points of intersection
Part D — Discriminant
7. For each pair, form the quadratic and state how many points of intersection there are (use the discriminant).
a $y=x^2$, $y=2x-1$
b $y=x^2+x+4$, $y=2x$
c $y=x^2-3x$, $y=-3$
d $y=x^2+2x+5$, $y=-2$
Challenge
8. The line $y=x+k$ is a tangent to the parabola $y=x^2$ (it touches at exactly one point). Find the value of $k$. (Hint: form the quadratic and set its discriminant to zero.)