Sketch and analyse linear and quadratic functions, finding gradient, intercepts, vertex and discriminant, and solving linear and quadratic equations and inequalities

What this dot point is asking

QCAA wants you to sketch and analyse linear and quadratic functions, find gradient, intercepts and vertex, and solve linear and quadratic equations and inequalities.

Linear functions

Form. $y = mx + c$. Gradient $m$, $y$-intercept $c$.

Gradient from two points. $m = (y_2 - y_1)/(x_2 - x_1)$.

Point-slope form. $y - y_1 = m(x - x_1)$.

**$x$-intercept.** $x = -c/m$.

Parallel and perpendicular. $m_1 = m_2$ for parallel; $m_1 m_2 = -1$ for perpendicular.

Inequalities. Solve as equations, but reverse the sign when multiplying or dividing by a negative.

Quadratic functions

Standard form. $y = ax^2 + bx + c$.

Factored form. $y = a(x - p)(x - q)$. Roots at $p$ and $q$.

Vertex form. $y = a(x - h)^2 + k$. Vertex at $(h, k)$.

Vertex from standard form. $x_v = -b/(2a)$. $y_v = c - b^2/(4a)$.

The discriminant

$\Delta = b^2 - 4ac$.

Quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Sketching a parabola

1. Identify $a$ (opens up if $a > 0$, down if $a < 0$).
2. Find $y$-intercept (substitute $x = 0$).
3. Find vertex via $-b/(2a)$.
4. Find $x$-intercepts (factor, complete the square, or use the formula).
5. Plot and draw a smooth parabola.

Worked example

Find the equation of the line through $(1, 4)$ and $(3, -2)$ in $y = mx + c$ form.

$m = (-2 - 4)/(3 - 1) = -6/2 = -3$.

Point-slope: $y - 4 = -3(x - 1)$. Expand: $y = -3x + 7$.

Common traps

Sign on $-b/(2a)$. Common slip is to drop the minus.

Vertex form sign of $h$. $(x - 3)^2$ has $h = 3$, not $-3$.

Confusing $a$ with leading coefficient. Always same in standard and vertex forms.

Quadratic-formula sign error. The denominator is $2a$ for the whole expression.

In one sentence

Linear functions $y = mx + c$ have gradient $m$ (parallel lines share $m$, perpendicular have $m_1 m_2 = -1$), and quadratics $y = ax^2 + bx + c$ have vertex at $x = -b/(2a)$ and discriminant $\Delta = b^2 - 4ac$ that classifies the roots ($> 0$ two real, $= 0$ one repeated, $< 0$ none).