Straight line (gradient-int, general, point gradient form): notes and practice questions

If a line L passes through point A(5,15) and is parallel to BC such that their coordinates are B(-2,1) and C (10,13).

(a) Find the equation of line L.

If line L passes through a point (c,-5)

(b) Find the value of c.

A kite PQRS is shown on the following set of axes.

The kite has vertices Q(1, 5) and S(7, -1) and is symmetrical about the diagonal [PR].

(i) Write down the coordinates of the midpoint of [QS].

(ii) Hence, find the equation of the line containing the diagonal [PR].

(b) Given that vertex P lies on the y-axis and the x-coordinate of R is 6, find the area of the kite PQRS.

Consider the function f defined by $f(x) = 50e^{-0.2x}$ for $x \in \mathbb{R}^+$.

The graph of f and the line $y = x$ intersect at point P.

Find the x-coordinate of P.

The line L has a gradient of -2 and is a tangent to the graph of f at the point Q.

Find the exact coordinates of Q.

Show that the equation of L is $y = -2x + 10 \ln 5 + 10$.

The shaded region A is enclosed by the graph of f and the lines $y = x$ and L.

Find the x-coordinate of the point where L intersects the line $y = x$.

Hence, find the area of A.

The line L is tangent to the graphs of both f and the inverse function $f^{-1}$.

Find the shaded area enclosed by the graphs of f and $f^{-1}$ and the line L.

The points P(1, 8), Q(–5, –2) and R(7, 4) are the vertices of a triangle.

(a) Find the equation of the altitude from vertex P to the side [QR].

(b) The altitude found in part (a) intersects the x-axis at the point D. Find the coordinates of D.

A function is defined by $f(x) = x^3 - 4x$.

(a) Find the equation of the tangent to the graph of $f$ at the point where $x = -1$.

(b) The tangent line found in part (a) intersects the graph of $f$ at a second point, P. Find the coordinates of P.

(c) Find the exact area of the finite region enclosed by the graph of $f$ and the tangent line.

A map is drawn on a Cartesian plane. A straight road connects two towns, Ashton, located at A(1, 7), and Brixton, located at B(9, -1).

(a) A service station is planned to be built at the midpoint of the road connecting the two towns. Find the coordinates of the service station.

A new road, represented by the line $L$, is to be built perpendicular to the road [AB]. This new road will pass through the service station.

(b) Find the gradient of the line $L$.

(c) Hence, write down the equation of the line $L$.

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 $m$.

A pharmaceutical company is testing a new drug. The concentration of the drug, $C$, in the bloodstream of a patient, in micrograms per millilitre ($\mu\text{g/mL}$), $t$ hours after administration, is modelled by the function $C(t) = 5te^{-0.2t}$, for $0 \le t \le 15$.

Sketch the graph of $C(t)$ for $0 \le t \le 15$, clearly indicating the coordinates of the initial concentration point $A$, the maximum concentration point $M$, and the concentration point $B$ at $t=15$ hours.

State the range of the concentration $C(t)$ during the observed period.

Find the equation of the straight line connecting the initial concentration point $A$ and the concentration point $B$ at $t=15$ hours.

Show that the rate of change of the drug concentration is given by $C'(t) = (5-t)e^{-0.2t}$.

At a certain time, the rate of change of the drug concentration is parallel to the line AB. Find the equation of the tangent line to the graph of $C(t)$ at this time. Give all coefficients in your equation correct to $3$ significant figures.

Calculate the area of the region enclosed by the graph of $C(t)$ and the line AB.

A landscape architect is designing a garden on a coordinate grid. Key points in the design are located at P(–3, 5), Q(1, 7), and S(0, -2).

(a) A straight path connects points P and Q. Find the equation of this path.

(b) A second path is to be laid parallel to the path PQ, passing through point S. Find the equation of the line for this second path.

(c) A third path is perpendicular to the path PQ and passes through point Q. Find the equation of this third path.

(d) A straight decorative fence is to be placed horizontally, passing through point P. Write down the equation of the line representing the fence.

The trajectory of a small drone flying over a landscape can be modelled by the function $f(x) = -0.5x^2 + 4x + 1$, where $x$ is the horizontal distance in meters from the launch point and $f(x)$ is the altitude in meters. The drone passes through a checkpoint A at a horizontal distance of 2 meters.

(a) (i) Find the gradient of the tangent to the drone's trajectory at checkpoint A.

(a) (ii) Hence, write down the gradient of the normal to the drone's trajectory at checkpoint A.

(b) Write down the equation of the normal to the drone's trajectory at checkpoint A.

(c) A searchlight beam is directed along the normal line found in part (b). This beam intersects the drone's trajectory again at a second point B. Find the coordinates of B.

The function $f$ is defined by $f(x) = e^x \sin(x)$, for $x \in [0, \pi]$.

(a) Show that the curve $y=f(x)$ has only one point of inflection in its domain, and determine its coordinates.

(b) Find the equations of the tangent and the normal to the curve at the point where $x = \pi$.

(c) Calculate the area of the triangle formed by this tangent, this normal, and the $y$-axis.

A civil engineer is designing a parabolic arch for a pedestrian bridge. The shape of the arch can be modelled by the function $f(x) = -x^2 + 7x - 5$, where $x$ is the horizontal distance in meters from one end of the bridge and $f(x)$ is the height of the arch above the ground in meters.

A support cable needs to be attached to the arch at a point A where $x=2$.

(i) Calculate the gradient of the tangent to the arch at point A.

(ii) Hence, write down the gradient of the line perpendicular to the tangent at point A.

The support cable is designed to be perpendicular to the arch at point A. Write down the equation of the line representing this support cable.

The support cable (represented by the normal line) is extended and intersects the parabolic arch again at a second point B. Find the coordinates of point B.

An architect is designing a decorative archway for a garden entrance. The shape of the archway's inner curve is modelled by the equation $y^2 = x^3 + ax + b$, where $x$ and $y$ are in meters.

(a.i) On the same set of axes, sketch the curve $C_1: y^2 = x^3$ for $x \ge 0$, clearly indicating any points of intersection with the coordinate axes. Assume a suitable range for $x$ and $y$ that shows the key features.

(a.ii) On the same set of axes, sketch the curve $C_2: y^2 = x^3 + 2x^2$ for $x \ge -2$, clearly indicating any points of intersection with the coordinate axes. Assume a suitable range for $x$ and $y$ that shows the key features.

(a.iii) By considering each curve from part (a), identify two key features that would distinguish $C_1$ from $C_2$.

(b.i) For the curve $C_2: y^2 = x^3 + 2x^2$, show that $\frac{dy}{dx} = \pm \frac{3x^2 + 4x}{2\sqrt{x^3 + 2x^2}}$ for $x > -2, x \ne 0$.

(b.ii) Find the $x$-coordinates of any local maximum or minimum points on $C_2: y^2 = x^3 + 2x^2$.

(c) The curve $C_2: y^2 = x^3 + 2x^2$ has points of inflexion. Find the $x$-coordinate of these points, giving your answer in the form $x = \frac{p \pm \sqrt{q}}{r}$ where $p, q, r \in \mathbb{Z}$.

Consider a different archway design modelled by the curve $C_3: y^2 = x^3 + 2$, for $x \ge -\sqrt[3]{2}$.

(d.i) The point P(-1, -1) is a rational point on $C_3$. Find the equation of the tangent to $C_3$ at P.

(d.ii) This tangent intersects $C_3$ at another rational point Q. Find the coordinates of Q, expressing each coordinate as a fraction.

(e) The point S(-1, 1) also lies on $C_3$. The line [QS] intersects $C_3$ at a further point R. Determine the coordinates of R.

A school is organizing a field trip to the local science museum. The cost to rent a bus for the day is $\$350$.

Additionally, there is an entry fee of $\$12.75$ per student.

(a) Write down a formula connecting the total cost of the trip ($T$) with the number of students attending ($s$).

(b) Explain why $T = f(s)$ is a function.

(c) Derive an expression for $s$ in terms of $T$.

The school has a maximum budget of $\$900$ for the field trip.

(d) Hence, calculate the greatest number of students that can attend the trip.

(e) Given that only $20$ students attend the trip, calculate how much each student should be charged so that the school covers its costs.

This question asks you to examine linear and quadratic functions constructed in systematic ways using geometric sequences.

Consider the function $L(x) = mx + c$ for $x \in \mathbb{R}$ where $m, c \in \mathbb{R}$ and $m, c \neq 0$.

Let $r \in \mathbb{R}$ be the root of $L(x) = 0$.

If $m, r$ and $c$, in that order, are in geometric sequence, then $L(x)$ is said to be a GS-linear function.

Show that $L(x) = 2x + 8$ is a GS-linear function.

Consider $L(x) = mx + c$.

Show that $r = -\frac{c}{m}$.

Given that $L(x)$ is a GS-linear function, show that $L(x) = mx + m^3$.

State any further restrictions on the value of $m$, assuming the common ratio $q \neq \pm 1$.

There are only two integer values of $m$ (excluding $m=0$) for which $L(x) = mx+m^3$ is a GS-linear function with a common ratio $q \neq \pm 1$. One of these gives $L(x) = 2x+8$.

Use part (b) to determine the other GS-linear function with integer values of $m, r$ and $c$ that satisfies the condition.

Consider the function $Q(x) = ax^2 + bx + c$ for $x \in \mathbb{R}$ where $a \in \mathbb{R}, a \neq 0$ and $b, c \in \mathbb{R}$.

Let $r_1, r_2 \in \mathbb{R}$ be the roots of $Q(x) = 0$.

Write down an expression for

(i) the sum of roots, $r_1 + r_2$, in terms of $a$ and $b$.

(ii) the product of roots, $r_1 r_2$, in terms of $a$ and $c$.

If $a, b, c$ are in arithmetic sequence, AND $r_1, r_2$ are in geometric sequence, then $Q(x)$ is said to be a GS-Quadratic function.

Given that $Q(x)$ is a GS-Quadratic function, show that $qr_1^2 + 2(1+q)r_1 + 1 = 0$, where $q = r_2/r_1$.

Hence or otherwise, show that $q=1$ or $q=-1$.

Consider the case where $q=1$.

Determine the two GS-Quadratic functions that satisfy this condition, given that $a=1$. Give your answers in the form $x^2 + Bx + C$.

Consider the case where $q=-1$.

Determine the two GS-Quadratic functions that satisfy this condition, given that $a=\pm 2$.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis.

(a) Write down the coordinates of the points where the arch meets the ground and the highest point of the arch.

(b) Find the equation of the parabolic arch.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(c) Express the width and height of the rectangular banner in terms of $x$.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(d) Write down an expression for the area of the rectangular banner, $A(x)$, in terms of $x$.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(e) Find the value of $x$ for which the area of the banner is maximized.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(f) Calculate the dimensions (width and height) of the banner that yield the maximum area.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(g) Find the maximum possible area of the inscribed rectangular banner.

The following diagram shows part of the graph of a quadratic function $f$ representing the path of a thrown ball.

The graph of $f$ has its vertex at $(2,5)$ and it passes through point $Q$ as shown.

The function can be written in the form $f(x)=a(x-h)^{2}+k$.

Write down the equation of the axis of symmetry.

(b) Write down the values of $h$ and $k$.

(c) Point $Q$ has coordinates $(4,13)$. Find the value of $a$.

(d) The line $L$ is tangent to the graph of $f$ at $Q$. Find the equation of $L$.

(e) Now consider another function $y=g(x)$. The derivative of $g$ is given by $g′(x)=f(x)-d$, where $d\in\mathbb{R}$. Find the values of $d$ for which $g$ is an increasing function.

(f) Find the values of $x$ for which the graph of $g$ is concave-up.

A landscape architect is designing a new garden. A straight path is to be built, and its boundary can be modelled by the equation $3x + 2y - 24 = 0$, where $x$ and $y$ are distances in metres.

(a) Write down the equation of the line in the form $y = mx + c$.

The garden's main feature is a triangular flower bed, with vertices at the origin O(0,0), and the points where the path intersects the $x$-axis (point A) and the $y$-axis (point B).

(b) Given that the line intersects the $x$-axis at point A and the $y$-axis at point B, find the coordinates of A and B.

A landscape architect is designing a new garden. A straight path is to be built, and its boundary can be modelled by the equation $3x + 2y - 24 = 0$, where $x$ and $y$ are distances in metres. The garden's main feature is a triangular flower bed, with vertices at the origin O(0,0), and the points where the path intersects the $x$-axis (point A) and the $y$-axis (point B).

(c) Calculate the area of triangle OAB.