maths1-week3

Week map

Quadratic notation

• $f(x) = ax^2 + bx + c$ → standard quadratic → $a \neq 0$ → $f(x) = 2x^2 - 4x + 1$.
• $a$ → leading coefficient → sign sets cup up ($a>0$) or down ($a<0$).
• Root / zero → input where $f(x)=0$ → solve $ax^2+bx+c=0$.
• $h = -\frac{b}{2a}$ → $x$-coordinate of vertex → axis of symmetry vertical line $x=h$.
• $D = b^2 - 4ac$ → discriminant → classifies real roots.

Vertex form

Roots and the discriminant

Factoring strategy (monic $a=1$)

Completing the square sketch

Graph features

• Axis of symmetry: $x = h$.
• $y$-intercept: $f(0) = c$.
• Direction: min value at vertex if $a>0$; max if $a<0$.
• Width: larger $|a|$ → narrower parabola.

Pattern families

Easy — Factor and solve

• Monic quadratics with integer roots.
• Read vertex from $(x-h)^2+k$ when given.
• State $y$-intercept from $c$.

Medium — Vertex and transformation

• Convert standard to vertex form by completing square or formula for $h,k$.
• Given roots, write quadratic in factored form $a(x-r_1)(x-r_2)$.
• Use symmetry: roots equidistant from axis at $h$.

Hard — Discriminant parameters

• Find values of $k$ so $x^2 + kx + 9 = 0$ has exactly one real root → $D=0$.
• Inequalities on $D$ for “no real roots” or “two real roots”.
• Word constraints linking vertex height to domain (e.g. minimum cost).

Worked mini-examples

Traps

• Sign error in $h = -b/(2a)$ when $b$ negative.
• Vertex $(h,k)$ confused with roots (zeros on $x$-axis).
• Forgetting $a$ when using formula: divide by $2a$, not 2.
• $D=0$ means one repeated root, not “no roots”.
• Opening direction from sign of $a$, not $c$.

Diagnostic (try yourself)

1. Solve $x^2 + 7x + 10 = 0$ by factoring.
2. Find the vertex and axis of symmetry of $f(x) = 2x^2 - 12x + 5$.
3. For what values of $m$ does $x^2 + mx + 16 = 0$ have exactly one real solution?
4. A parabola has roots at $x = 1$ and $x = 9$ and passes through $(0, 18)$. Write $f(x)$ in standard form.
5. Without fully solving, how many real roots does $2x^2 + 3x - 7 = 0$ have? Use the discriminant.