Quadratic functions (general form, roots form, vertex form – intercepts, symmetry): notes and practice questions

Consider the function $g(x) = -2(x + 1)(x - p)$, where $x \in \mathbb{R}$ and $p$ is a real constant.

The axis of symmetry of the graph of $g$ has equation $x = 3$.

Show that $p = 7$.

Find the coordinates of the vertex of the graph of $g$.

Hence, write down the range of $g$.

A scientist is modeling the behavior of a particle in a fluctuating electromagnetic field. The particle's position over time can be described by a quadratic equation involving a field strength parameter $p$. The equation is given by $px^2 - (p + 7)x + 4p + 28 = 0$, where $p \in \mathbb{R}$.

(a) Write down an expression for the product of the roots of this equation, in terms of $p$.

(b) Hence or otherwise, determine the values of $p$ such that the equation has one positive and one negative real root.

A drone takes off from a platform. Its height, $h$ metres, above the platform after $t$ seconds is given by $h(t) = 6t - t^2$, for $0 \le t \le 8$. This is shown in the following diagram.

The drone lands back on the platform when $t=p$.

Find the value of $p$.

The drone reaches its maximum height when $t=q$.

Find the value of $q$.

Find the drone's maximum height above the platform.

Find the drone's vertical distance from the platform when $t=8$.

The total vertical distance travelled by the drone in the first 8 seconds is given by $d$.

Find the value of $d$.

A second drone, Drone B, takes off from the same platform. Its velocity is given by $v_B(t) = 8 - 2t$, for $t \ge 0$.

When $t = k$, the total vertical distance travelled by Drone B is equal to $d$.

Find the value of $k$.

Let $g(x) = 3 + 2x - x^2$. The function can be written in the form $g(x) = k - (x-h)^2$.

(i) Write down the value of $h$.

(ii) Find the value of $k$.

A function is defined by $f(x) = x^2 - 6x + c$, where $x, c \in \mathbb{R}$. The graph of $f$ is tangent to the line $L$ with equation $y = 2x - 11$.

(a) Show that $c=5$.

(b) The function $f$ can be expressed in the form $f(x) = (x-p)(x-q)$, where $p, q \in \mathbb{R}$.

Find the value of $p$ and the value of $q$.

(c) The function $f$ can also be expressed in the form $f(x) = (x-h)^2 + k$, where $h, k \in \mathbb{R}$.

Find the value of $h$ and the value of $k$.

(d) Hence find the values of $x$ where the graph of $f$ is both positive and decreasing.

A function is defined by $f(x) = \frac{1}{2}x^2 + x + 4$. The following diagram shows part of the graph of $f$.

The graph has a vertex at V and intersects the y-axis at point P.

(a) Find the coordinates of the vertex V.

(b) Write down the coordinates of the y-intercept, P.

(c) The line L is the normal to the graph of $f$ at point P. Find the equation of L, giving your answer in the form $y=mx+c$.

(d) The line L intersects the graph of $f$ at a second point, Q. Calculate the distance between P and Q.

A function, $g$, has its derivative given by $g'(x) = -2x^2 + 8x + k$, where $k \in \mathbb{R}$. The following diagram shows part of the graph of $g'$.

The graph of $g'$ has an axis of symmetry $x = q$.

(a) Find the value of $q$.

(b) The vertex of the graph of $g'$ has a y-coordinate of 10. Find the value of $k$.

(c) Find the equation of the tangent to the graph of $g'$ at $x = 0$.

The graph of $g$ has a point of inflexion at $x = c$.

(d) (i) Find the value of $c$.

(ii) Find the values of $x$ for which the graph of $g$ is concave-up. Justify your answer.

A team of engineers is designing a new roller coaster ride. The path of a certain section of the ride can be modelled by the function $g(x)=\frac{x^2 + 3x - 10}{x-4}$, where $x$ is the horizontal distance in metres from the starting point and $g(x)$ is the vertical height in metres. The domain of the function is $x \in \mathbb{R}, x\neq 4$.

Find the coordinates where the path of the roller coaster crosses the horizontal ground (x-axis).

Find the coordinates where the path of the roller coaster crosses the vertical axis (y-axis).

Write down the equation of the vertical asymptote of the graph of $g$.

The oblique asymptote of the graph of $g$ can be written as $y = ax + b$ where $a, b \in \mathbb{Z}$.

Find the value of $a$ and the value of $b$.

Sketch the graph of $g$ for $-20 \le x \le 20$, clearly indicating the points of intersection with each axis and any asymptotes.

Engineers want to analyse the inverse of the roller coaster's height function, $h(x) = \frac{1}{g(x)}$.

Express $h(x)$ in partial fractions.

Hence find the exact value of $\int_{0}^{1} h(x) dx$, expressing your answer as a single logarithm.

Consider the functions $f(x) = 5 - x^2$, $g(x) = \frac{3}{x-2}$, and $h(x) = e^x + 1$.

(a) Find the range of $f(x)$.

(b) Find the range of $g(x)$.

(c) Find the range of $h(x)$.

(d) Find an expression for $(g \circ f)(x)$.

(e) Solve the equation $(g \circ f)(x) = -1$.

(f) Solve the inequality $(h \circ f)(x) < e+1$.

A student claims that $2^n > n^2$ for all integers $n \ge 5$.

(a) Show that $2n^2 > (n+1)^2$ for all integers $n \ge 3$.

(b) Use mathematical induction and the result from part (a) to prove that the student's claim is valid for all integers $n \ge 5$.

Consider the function $g(x) = (m-1)x^2 - 4mx + 5m$, for $x \in \mathbb{R}$ and $m \in \mathbb{R}, m \neq 1$.

(a) Find the range of values for $m$ for which the equation $g(x)=0$ has two distinct real roots.

(b) Find the possible values of $m$ for which the vertex of the graph of $y=g(x)$ lies on the line $y=x$.

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ by

$f(x) = 2x^2 + 8x + k$

$g(x) = -x^2 + mx + 1$

where $k$ and $m$ are constants.

The vertex of the graph of $y=f(x)$ and the vertex of the graph of $y=g(x)$ both lie on the line with equation $y = -x + 3$.

(a) Show that $k=13$.

(b) Given that $m > 0$, find the value of $m$.

This question explores the characteristics of a company's profit function, $P_n(x) = x^n(L-x)^n$, where $x$ represents the number of units produced (in thousands), $L$ is the maximum production capacity (in thousands of units), with $L \in \mathbb{R}^+$ and $n \in \mathbb{Z}^+$.

For parts (a) and (b), consider the case where $L = 4$.

Consider $P_1(x) = x(4-x)$.

Sketch the graph of $y = P_1(x)$, clearly indicating the values of any axes intercepts and the coordinates of any local maximum or minimum points.

Consider $P_n(x) = x^n(4-x)^n$, where $n \in \mathbb{Z}^+$, $n > 1$.

Use your graphic display calculator to explore the graph of $y = P_n(x)$ for:

• the odd values $n = 3$ and $n = 5$;
• the even values $n = 2$ and $n = 4$.

Hence, copy and complete the following table:

Number of local maximum points | Number of local minimum points | Number of points of inflexion with zero gradient

---|---|---

$n=3$ and $n = 5$ | |

$n=2$ and $n = 4$ | |

Now consider $P_n(x) = x^n(L-x)^n$ where $L \in \mathbb{R}^+$ and $n \in \mathbb{Z}^+$, $n > 1$.

Show that $P_n'(x) = nx^{n-1}(L-2x)(L-x)^{n-1}$.

State the three solutions to the equation $P_n'(x) = 0$.

Show that the point $(\frac{L}{2}, P_n(\frac{L}{2}) )$ on the graph of $y = P_n(x)$ is always above the horizontal axis.

Hence, or otherwise, show that $P_n'(\frac{L}{4}) > 0$, for $n \in \mathbb{Z}^+$.

By using the result from part (f) and considering the sign of $P_n'(-1)$, show that the point $(0, 0)$ on the graph of $y = P_n(x)$ is

(i) a local minimum point for even values of $n$, where $n > 1$ and $L \in \mathbb{R}^+$;

(ii) a point of inflexion with zero gradient for odd values of $n$, where $n > 1$ and $L \in \mathbb{R}^+$.

Consider the graph of $y = x^n(L-x)^n - k$, where $n \in \mathbb{Z}^+$, $L \in \mathbb{R}^+$ and $k \in \mathbb{R}$.

State the conditions on $n$ and $k$ such that the equation $x^n(L-x)^n = k$ has four distinct solutions for $x$.

A tech company's daily profit, $P$, in thousands of dollars, from producing $x$ units of a new gadget is modelled by the function $f(x) = -2x^2 + 16x + 468$, for $x \in \mathbb{R}$.

(a) Find the range of the company's daily profit.

Due to new environmental regulations, the company faces a levy that adjusts its profit. The adjusted profit, $A$, is given by the function $g(P) = P + k$, where $k \in \mathbb{R}$ is a constant representing the levy's impact.

Given that the adjusted profit $(g \circ f)(x)$ must be non-positive for all $x \in \mathbb{R}$, determine the set of possible values for $k$.

A civil engineer is designing a section of a curved bridge support. The vertical profile of the support can be modelled by the function $h(x) = x^3 + ax^2 + b$, where $h$ is the height above the ground in meters and $x$ is the horizontal distance in meters from a reference point. The ground level is represented by $h=0$.

(a) Consider the case where $a = -6$.

(i) Determine the two values of $b$ such that the bridge support profile $h(x) = x^3 - 6x^2 + b$ touches the ground at exactly two points (i.e., has exactly two $x$-axis intercepts).

(ii) State the set of values of $b$ for which the profile has exactly one $x$-axis intercept.

(iii) State the set of values of $b$ for which the profile has exactly three $x$-axis intercepts.

(b) Show that the critical points (points of zero gradient) of the function $h(x) = x^3 + ax^2 + b$ are located at $P(0, b)$ and $Q\left(-\frac{2}{3}a, \frac{4}{27}a^3+b\right)$.

(c) For $a > 0$, determine whether $P$ is a local maximum or minimum, and whether $Q$ is a local maximum or minimum.

(d) Prove that the bridge support profile $h(x) = x^3 + ax^2 + b$ has exactly three points where it meets the ground (three $x$-axis intercepts) if and only if $4a^3b + 27b^2 < 0$.

The population, $P$, in hundreds of plants, $x$ years after the study began, is modeled by the function $f(x) = -2x^2 + 8x + 5$, for $x \in \mathbb{R}$.

(a) Find the range of $f$.

Due to environmental changes, a new invasive plant species is introduced. The impact of this invasive species on the orchid's population is modeled by a function $g(P) = -P + k$, where $P$ is the current orchid population (in hundreds of plants) and $k$ is a constant representing the environmental resilience. For the orchid species to survive and thrive, the combined effect $(g \circ f)(x)$ must always be non-negative (i.e., $(g \circ f)(x) \ge 0$) for all $x \in \mathbb{R}$. Determine the set of possible values for $k$.

A pharmaceutical company is formulating a new drug. They are testing two active ingredients, $x_1$ and $x_2$, such that their total concentration is $24$ mg/mL. The drug's efficacy is modelled by the product of the concentrations of the two ingredients.

Find the efficacy, $E$, as a function of $x_1$ only.

Determine the concentration of $x_1$ that maximizes the drug's efficacy.

Hence, show that the maximum efficacy for two ingredients with a total concentration of $24$ mg/mL is $144$.

Let $M_n(S)$ represent the maximum efficacy for a drug with $n$ active ingredients and a total concentration of $S$ mg/mL. For $n = 2$, the maximum efficacy can be expressed as $M_2(S) = \left(\frac{S}{2}\right)^2$.

Verify that $M_2(S) = \left(\frac{S}{2}\right)^2$ is true for $S = 24$.

The relationship between the geometric mean and arithmetic mean states that for $n$ positive real numbers $x_1, x_2, ..., x_n$, their geometric mean $(x_1 \times x_2 \times ... \times x_n)^{\frac{1}{n}}$ is always less than or equal to their arithmetic mean $\frac{x_1 + x_2 + ... + x_n}{n}$.

Show that the geometric mean and arithmetic mean are equal when $x_1 = x_2 = ... = x_n$.

Use this result to prove that $M_n(S) = \left(\frac{S}{n}\right)^n$.

Using the formula for $M_n(S)$, determine the value of:

$M_3(24)$;

$M_4(24)$;

$M_5(24)$.

For a fixed total concentration of $S = 24$ mg/mL, the company wants to find the optimal number of active ingredients, $n$, to maximize the drug's efficacy. Let $P(S)$ denote this maximum efficacy.

Write down the value of $P(24)$ and the value of $n$ at which it occurs.

Determine the value of $P(30)$ and the value of $n$ at which it occurs.

Consider the continuous function $h$, defined by $\ln(h(x) ) = x\ln\left(\frac{S}{x}\right)$, where $x \in \mathbb{R}^+$. A sketch of the graph of $y = h(x)$ is shown in the following diagram. Point A is the maximum point on this graph.

Find, in terms of $S$, the $x$-coordinate of point A.

Verify that $h(x) = M_x(S)$, when $x \in \mathbb{Z}^+$.

The company has a total concentration of $S = 150$ mg/mL available. Use your answer to part (h) to find the largest possible efficacy. Give your answer in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}^+$.

The total number of units produced, $P$, by a factory depends on the number of hours, $H$, the factory operates. A production manager uses the model $P = -0.8H^2 + 28H + 50$ to predict the total units produced on any given day, where $5 \le H \le 20$.

An energy auditor investigates the relationship between the total units produced and the energy consumption, $E$, in kilowatt-hours (kWh). The following table shows the data collected on five different days.

$\begin{array}{|c|c|} \hline \textbf{Units Produced (P)} & \textbf{Energy Consumption (E, in kWh)} \\ \hline 225 & 16.2 \\ 250 & 17.4 \\ 275 & 18.8 \\ 290 & 19.5 \\ 305 & 20.3 \\ \hline \end{array}$

Use the production model to estimate the number of units produced when the factory operates for 15 hours.

Find an appropriate regression equation that will allow the auditor to predict the energy consumption on a day when $P$ units are produced.

Hence, use your regression equation to predict the energy consumption when the factory operates for 15 hours.

A quadratic function $h(x)$ has its axis of symmetry at $x=-1$. One of its x-intercepts is at $x=2$.

(a) Find the other x-intercept.

The graph of $h(x)$ passes through the point $(0, 8)$.

(b) Find the equation of $h(x)$, giving your answer in the form $y=Ax^2+Bx+C$.

A line $L$ with equation $y = mx + 12$ is a tangent to the graph of $h(x)$.

(c) Find the possible values of $m$.