Exercises: Cubic Graphs and Graphical Solutions
Let $f(x) = x^3 - 3x$. Copy and complete the table of values for $x = -2, -1, 0, 1, 2$.
Let $f(x) = x^3 - 3x$. Copy and complete the table of values for $x = -2, -1, 0, 1, 2$.
The graph of $f(x) = x^3 - 4x$ crosses the $x$-axis at $x = -2$, $x = 0$ and $x = 2$. a) Write down the solutions of $f(x) = 0$. b) Write down the coordinates where the…
The graph of $f(x) = x^3 - 4x$ crosses the $x$-axis at $x = -2$, $x = 0$ and $x = 2$.
- a) Write down the solutions of $f(x) = 0$.
- b) Write down the coordinates where the graph crosses the $y$-axis.
For $f(x) = x^3 - 3x + 1$, the graph passes through $(-2,-1)$, $(-1,3)$, $(0,1)$, $(1,-1)$ and $(2,3)$. Using these points, state which two consecutive integers each…
For $f(x) = x^3 - 3x + 1$, the graph passes through $(-2,-1)$, $(-1,3)$, $(0,1)$, $(1,-1)$ and $(2,3)$.
Using these points, state which two consecutive integers each root of $f(x) = 0$ lies between.
Let $f(x) = x^3 - 6x^2 + 9x$ for $0 \le x \le 4$. a) Complete a table of values for $x = 0, 1, 2, 3, 4$. b) Draw the graph of $f$ for $0 \le x \le 4$. c) Use your graph…
Let $f(x) = x^3 - 6x^2 + 9x$ for $0 \le x \le 4$.
- a) Complete a table of values for $x = 0, 1, 2, 3, 4$.
- b) Draw the graph of $f$ for $0 \le x \le 4$.
- c) Use your graph to estimate the solutions of $f(x) = 2$.
Let $f(x) = x^3 - 2x^2 - 5x + 6$ for $-3 \le x \le 4$. a) Complete a table of values for integer values of $x$ from $-3$ to $4$. b) Draw the graph of $f$. c) Use the…
Let $f(x) = x^3 - 2x^2 - 5x + 6$ for $-3 \le x \le 4$.
- a) Complete a table of values for integer values of $x$ from $-3$ to $4$.
- b) Draw the graph of $f$.
- c) Use the graph to write down the solutions of $f(x) = 0$.
Let $f(x) = x^3 - 3x$ for $-2 \le x \le 2$. On the same axes, draw the graph of $g(x) = x + 1$. a) Complete a table of values for both functions. b) Draw both graphs. c)…
Let $f(x) = x^3 - 3x$ for $-2 \le x \le 2$. On the same axes, draw the graph of $g(x) = x + 1$.
- a) Complete a table of values for both functions.
- b) Draw both graphs.
- c) Use your graphs to estimate the solutions of $x^3 - 3x = x + 1$ for $-2 \le x \le 2$.
Let $f(x) = -x^3 + 3x^2 + 4$ for $-2 \le x \le 4$. a) Draw the graph of $f$ using integer values of $x$. b) Use your graph to estimate the solution of $f(x) = 0$. c)…
Let $f(x) = -x^3 + 3x^2 + 4$ for $-2 \le x \le 4$.
- a) Draw the graph of $f$ using integer values of $x$.
- b) Use your graph to estimate the solution of $f(x) = 0$.
- c) Find the range of values of $k$ for which $f(x) = k$ has three distinct real solutions, using $f'(x)$ to find the turning points.
Let $f(x) = x^3 - 7x + 6$. a) Show that $x = 1$ is a root of $f(x) = 0$. b) Factorise $f(x)$ fully and write down all the roots. c) Sketch the graph of $f$, marking the…
Let $f(x) = x^3 - 7x + 6$.
- a) Show that $x = 1$ is a root of $f(x) = 0$.
- b) Factorise $f(x)$ fully and write down all the roots.
- c) Sketch the graph of $f$, marking the intercepts with both axes.
A cubic function is $f(x) = x^3 + bx^2 + cx + d$. Its graph passes through $(0, 4)$, $(1, 0)$ and $(-1, 6)$. a) Find the values of $b$, $c$ and $d$. b) Show that $x = 1$…
A cubic function is $f(x) = x^3 + bx^2 + cx + d$. Its graph passes through $(0, 4)$, $(1, 0)$ and $(-1, 6)$.
- a) Find the values of $b$, $c$ and $d$.
- b) Show that $x = 1$ is a root and find the other roots of $f(x) = 0$, if any.
- c) Find the number of solutions of $f(x) = 4$.