Algebra and Graphs

A square has a side length of $a + b$. A smaller square of side length $b$ is removed from one corner, as shown. What is the area of the remaining L-shaped region in terms of $a$ and $b$?

Factorise the expression $x^2 - 10x + 25$ completely. The diagram illustrates the vertex of the corresponding parabola.

The diagram shows a rectangle with dimensions $2x + 1$ and $x - 4$. Express the area of the rectangle as a simplified quadratic expression.

Simplify the algebraic expression $\frac{x^2 - 9}{x + 3}$. The diagram shows the graph of the numerator function $y = x^2 - 9$.

Factorise the expression $2x^2 + 5x - 3$. The diagram shows the region where the quadratic function is negative.

Given that $(x - 1)$ is a factor of $f(x) = x^3 + kx - 2$, find the value of $k$. The diagram shows the point $(1, 0)$ where the curve $y = f(x)$ crosses the x-axis.

The diagram shows the intersection of the line $y = x + 2$ and the curve $y = x^2$. Find the coordinates of the intersection points by solving $x^2 - x - 2 = 0$.

The height of a projectile is modeled by $H(t) = -5t^2 + 20t + 25$. Factorise this expression completely to find the time $t$ when the projectile hits the ground (where $H(t) = 0$ and $t > 0$).

A rectangular block of marble has a volume of $x^3 + 5x^2 - 4x - 20$. If two of its dimensions are $(x+5)$ and $(x-2)$, what is the third dimension?

In a manufacturing process, the efficiency $E$ is given by $E = \frac{4x^2 - 25}{2x + 5}$. If $x$ represents the number of automated machines, simplify the expression for $E$.

For the function $f(x) = (x-1)(x+3)$, find the $x$-coordinate of the axis of symmetry.

Solve the linear inequality $2x + 3 < 11$ represented on the number line.

Given the linear graph of $y = mx + c$, if the line passes through $(0, 5)$ and $(2, 9)$, what is the value of $m$?

Find the roots of the quadratic equation $x^2 - 5x + 6 = 0$ as seen where the graph crosses the x-axis.

The line $y = kx - 7$ and the curve $y = x^2 - 4x + 2$ do not intersect. Find the range of values for $k$.

The diagram represents the graph of $y = ax^2 + bx + c$. Given the coordinates of the vertex and the y-intercept shown, find the equation of the quadratic.

The diagram shows the intersection of the line $y = 2x + c$ and the curve $y = x^2 - 3x + 4$. If the line is a tangent to the curve, determine the value of the constant $c$.

A cooling system's efficiency $E$ depends on temperature $T$ according to $E = -T^2 + 40T + 500$. A technician finds that for the system to remain stable, $E$ must be at least 800 units. Determine the range of temperatures $T$ for stable operation.

A gardener uses 60m of fencing to enclose a rectangular area against a straight stone wall (no fencing needed for the wall side). If the width perpendicular to the wall is $x$, find the maximum area possible.

The demand for a product is $q = 100 - 2p$ and the supply is $q = 0.5p^2 - 10$. Find the equilibrium price $p$ where demand equals supply (for $p > 0$).

A rectangle has a perimeter of 20 cm and its length $l$ is 2 cm more than its width $w$. The diagram illustrates the relationship between the dimensions. Solve the simultaneous equations $2l + 2w = 20$ and $l = w + 2$ to find the area of the rectangle.

The diagram shows a line with the equation $y = x + 3$ and a curve with the equation $y = x^2 - 3x + 6$. Find the coordinates of the points of intersection of the line and the curve.

Solve the simultaneous equations $x + y = 5$ and $x^2 + y^2 = 13$ represented by the line and circle in the diagram. One solution is $(3, 2)$. What is the other solution?

The diagram shows the intersection of the line $y = 3$ and the curve $y = \frac{12}{x}$. Find the value of $x$ at the intersection point $R$.

The diagram shows the line $y = 2x - 5$ and the circle $x^2 + y^2 = 25$. The line intersects the circle at two points $A$ and $B$. Find the $x$-coordinates of these two points.

The curve $y = x^2 - kx + 4$ is always above the $x$-axis. What is the range of $k$?

Find the point(s) of intersection of the curves $y = \frac{6}{x}$ and $y = 7 - x$.

An enclosure is made of a rectangular area and a semi-circular area attached to one of its sides of length $x$. The total perimeter of the enclosure is $P = x + 2y + \frac{\pi x}{2} = 20$ meters. If the total area $A = xy + \frac{\pi x^2}{8}$ is exactly $25$ square meters, find the value of $x$ that satisfies these simultaneous conditions. (Use $\pi \approx 3.14$ for estimation if needed, but solve algebraically).

The production cost $C$ (in thousands of dollars) of an item is modeled by the function $C = 2x^2 - 12x + 25$, where $x$ is the quantity produced in hundreds. The revenue $R$ is modeled by the linear function $R = 4x + 9$. At what production levels $x$ does the company break even (where $C = R$)?

A straight line with equation $y = kx - 5$ is a tangent to the curve $y = x^2 - 4x - 1$. By solving the simultaneous equations and using the discriminant, find the two possible values of the constant $k$.

An inequality is given by $y < 3$. What does the boundary line look like?

Which of these points satisfies the inequality $2x + y > 5$?

Solve $4(x - 2) \geq 12$.

A car rental costs $20 plus$0.50 per mile ($m$). If a customer wants to spend no more than $100, which inequality is correct?

In the linear programming problem, the feasible region is a quadrilateral with vertices $(0,0), (4,0), (2,4), (0,5)$. Which vertex maximizes $Z = 5x + 2y$?

For what values of $k$ does the point $(k, 2k)$ satisfy $3x + y < 10$?

Determine the inequality for a region that lies to the left of $x = 4$ and above $y = x - 2$.

An investor allocates funds between high-risk stocks ($x$) and low-risk bonds ($y$). The total investment is at most $10,000$. The investor requires the amount in bonds to be at least $3,000$ and at least as much as the amount in stocks. Which of the following defines the possible values of $x$?

A bakery makes chocolate cakes and vanilla cakes. A chocolate cake requires $300$g of flour and $100$g of sugar. A vanilla cake requires $200$g of flour and $150$g of sugar. The bakery has $6$kg of flour and $3$kg of sugar available. Let $x$ be chocolate and $y$ be vanilla cakes. If the bakery wants to maximize the total number of cakes, which of these constraints is the most restrictive at the point where they produce equal numbers of both?

A theater has $500$ seats divided into 'Standard' and 'Premium'. Standard seats sell for $20$ and Premium for $50$. The theater must sell at least $300$ tickets in total to cover costs. To maintain exclusivity, the number of Premium seats sold cannot exceed half the number of Standard seats sold. If $x$ is Standard and $y$ is Premium, which point $(x, y)$ represents a valid ticket sale?

Consider the function $h(x) = \sqrt{x-2}$ for $x \ge 2$. Find the value of $h^{-1}(3)$.

The diagram shows the mapping of values from set $A$ to set $B$ under the function $g(x) = x^2 + 1$. If $g(a) = 10$, find the positive value of $a$.

The function $f$ is defined by $f(x) = 2x - 5$ for $x \in \mathbb{R}$. Find the inverse function $f^{-1}(x)$.

Which of the following describes the relationship between the graph of $f$ and $f^{-1}$?

The graph of $y = f(x)$ is shown. The function is defined by $f(x) = \frac{a}{x+b}$. Using the vertical and horizontal asymptotes shown, find $f^{-1}(x)$.

The functions $f$ and $g$ are defined by $f(x) = 3x + 2$ and $g(x) = \frac{1}{x-1}$ for $x \neq 1$. Solve the equation $f(g(x)) = 5$.

The function $f$ is defined as $f(x) = x^2 - 4x + 7$ for the domain $x \ge k$. Given that $f$ has an inverse, what is the smallest possible value of $k$?

A function $f$ is defined by $f(x) = x^2 + bx + c$. The diagram shows the graph of $y=f(x)$ where the minimum point is at $(2, -1)$. Determine the expression for $f^{-1}(x)$ for the domain $x \ge 2$.

Given the function $f(x) = \frac{1}{x+1}$ for $x > -1$, the graph of $y=f(x)$ is shown. If a second function $g(x) = 2x - 3$ is introduced, find the value of $x$ for which $f(g(x)) = \frac{1}{4}$.

Two functions are defined as $f(x) = 3x - k$ and $g(x) = \frac{x+5}{2}$. If the composite function $f(g(x))$ passes through the point $(1, 4)$, find the value of the constant $k$.

What is the value of $y$ when $x = 0$ for the function $y = 2^{x+3}$?

Which of these is the graph of $y = -x^2 + 1$?

A line passes through $(0, -2)$ and $(4, 0)$. Determine the value of its $y$-intercept.

Find the gradient of a line perpendicular to the line $y = \frac{1}{4}x - 2$.

For what value of $k$ does the line $y = 4x + k$ touch the parabola $y = x^2$ at exactly one point?

Find the distance between the points of intersection of $y = x^2 - 4$ and the $x$-axis.

The curve $y = 3^x$ is transformed into $y = 3^{x-2}$. Describe the transformation.

The radioactive decay of an isotope is given by $N(t) = N_0(0.5)^{t/h}$, where $h$ is the half-life. If it takes $15$ days for the sample to decay to $20\%$ of its original mass, what is the half-life $h$ in days?

A rectangle has a perimeter of $100$ meters. If one side is $x$, the area is given by $A = x(50 - x)$. An architect wants to ensure the area is at least $600$ square meters. What is the range of possible values for $x$?

A power line is suspended between two poles $20$ meters apart. The shape of the cable is modeled by $y = 0.1x^2 - 2x + 15$ for $0 \le x \le 20$. Determine the minimum height of the cable above the ground and its horizontal position.

The graph of $y = 10/x$ is given. To solve the equation $10/x = x^2$, we look for the intersection with which curve?

The diagram shows $y = x^3 - 3x^2 + 2$. By looking at the graph, how many roots does the equation $x^3 - 3x^2 + 2 = 0$ have?

Given the graph of $y = x^2 - 3x$, what is the equation of the line that should be drawn to solve $x^2 - 3x - 2 = 0$?

To solve $x^2 - 5x + 4 = 0$ using the graph of $y = x^2 - 5x$, what line should be drawn?

The diagram shows the intersection of $y = x^3$ and $y = x + 1$. This intersection represents the solution to which equation?

A curve has equation $y = x^2 - 2x$. If the line $y = m$ intersects the curve at exactly one point, what is the value of $m$?

To solve $x + \frac{1}{x} = 3$ using the graph of $y = \frac{1}{x}$, which line should be drawn?

A structural engineer is modeling the stress on a beam. The vertical displacement $y$ of a specific point relative to a load position $x$ is given by the curve $y = x^3 - 3x^2 + 2$. To find the equilibrium positions under a specific external force, the engineer needs to solve the equation $x^3 - 3x^2 - x + 5 = 0$ using a graphical method. Given the graph of $y = x^3 - 3x^2 + 2$, what is the equation of the straight line that should be drawn to find the real roots of the equilibrium equation?

Two drones travel on paths defined by $y = \frac{1}{x^2}$ and $y = x + 1$ for $x > 0$. Using the provided graph, find the approximate x-coordinate where their paths cross.

A hydraulic system is governed by $y = |x - 3|$ and $y = 0.5x^2 - 2$. Find the positive x-coordinate where these two components reach an equilibrium as shown on the graph.

A curve has the equation $y = x^2 - 4x + 5$. The diagram shows the graph of this function. Find the gradient of the curve at the point $P(3, 2)$.

Differentiate $f(x) = 2x^4 - 6x$ and find $f'(1)$.

The gradient of the curve $y = kx^2$ at $x=1$ is $10$. Find the value of $k$.

Find the derivative of $y = \frac{1}{3}x^3$.