11.4 Quadratic Functions

Solve $x^2 - 9x + 20 = 0$ by factorising.

Solve $x^2 - 25 = 0$.

Solve $x^2 + 5x + 6 = 0$ using factorising.

State the coordinates of the vertex of $y = (x - -2)^2 + 5$ and whether it is a min or max.

State the $y$-intercept of $y = x^2 - -3x + 4$.

Solve $2x^2 - 5x - 3 = 0$ by factorising.

State the axis of symmetry of $y = x^2 - 4x + -1$.

Solve $x^2 - 4x - 5 = 0$ using the quadratic formula.

Does $y = -2x^2 + 3x + 1$ open upwards or downwards?

State the $x$-intercepts of $y = (x - 4)(x + 3)$.

Express $x^2 + 6x + 5$ in the form $(x + p)^2 + q$.

Find the vertex of $y = x^2 - 6x + 11$.

Solve $2x^2 - x - 6 = 0$ by factorising.

Expand $(x - 3)(x + 5)$ and write in $ax^2 + bx + c$ form.

Write $y = 2(x - 3)^2 - 5$ in expanded form $y = ax^2 + bx + c$.

Solve $x^2 + 8x + 10 = 0$ exactly, leaving any irrational answer in surd form.

A number plus its square is 30. Find the number.

For $y = (x - 3)^2 - 4$, state (a) the vertex, (b) the $x$-intercepts.

Express $y = 2x^2 - 8x + 5$ in vertex form.

A rectangle has length $(x + 3)$ cm and width $x$ cm. Its area is 18 cm$^2$. Find $x$.

Express $f(x) = x^2 + 8x + 10$ in vertex form, then state the range of $f$.

A quadratic $y = x^2 + bx + 7$ has its minimum value at $x = 3$. Find $b$.

How many real roots does $2x^2 - 4x + 3 = 0$ have? Use the discriminant.

The line $y = 2x + 1$ is tangent to the curve $y = x^2 + k$. Find $k$.

Solve $x^2 - 5x + 6 \leq 0$.

A monic quadratic has roots $-2$ and $5$. Write it in expanded form.

A ball thrown follows $h(t) = -5t^2 + 20t + 1$ ($h$ in m, $t$ in s). Find (a) the maximum height, (b) the time it occurs.

For $x^2 + 5x - 6 = 0$, find the sum and product of the roots without solving.

A fountain jet has $h(x) = -(x - 4)^2 + 9$ ($h$ height in m, $x$ horizontal distance in m). Find (a) max height, (b) range of $x$ above ground.

For what range of $k$ does $x^2 + kx + 9 = 0$ have no real roots?

A rectangle has perimeter 26 cm and area 40 cm$^2$. Find its dimensions.

For what values of $k$ does the line $y = x + k$ intersect $y = x^2$ at two distinct points?

A quadratic $y = ax^2 + bx + c$ passes through $(0, 5)$, $(1, 8)$ and $(3, 2)$. Find $a$, $b$, $c$.

Express $y = x^2 - 6x + 11$ in vertex form. Hence describe the transformation from $y = x^2$ to $y = x^2 - 6x + 11$.

A parabola has vertex $(2, 3)$ and passes through $(5, 30)$. Find its equation.

Find the points of intersection of $y = x^2 + 2x - 3$ and $y = x + 1$.

Find the minimum value of $f(x) = 3x^2 - 12x + 7$.

Show that for all real $k$, the equation $x^2 + 2kx + (k^2 + 1) = 0$ has no real solutions.

A farmer has 40 m of fencing to enclose a rectangular pen against a wall (so one side is the wall). Find the dimensions giving the maximum area.

A ball is thrown and its height (m) at times $t = 0, 1, 2$ is recorded as $1.5, 5, 7.5$ m. Assuming $h(t) = at^2 + bt + c$, find $a$, $b$, $c$ and predict $h(3)$.