internet intelligent tutors
Stretch Rule 1 For $y=pf(x)$, $p \gt 0$, the effect of $p$ is to vertically stretch the graph by a factor by $p$. Stretch Rule 2 For $y=f(kx)$, $k \gt 0$, the effect of $k$ is to horizontally compress the graph by a factor of $k$. Example 1 Given that the point $(-2,1)$ lies on […]
For $y=-f(x)$, we reflect $y=f(x)$ in the $x$-axis. For $y=f(-x)$, we reflect $y=f(x)$ in the $y$-axis. Example 1 Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of $y=f(x)$ and $y=-f(x)$. \( \begin{align} \displaystylef(x) &= x^3-4x^2+4x \\&= x(x^2-4x + 4) \\&= x(x-2)^2\end{align} \) Example 2 Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of […]
Rational functions are characterised by the presence of both a horizontal asymptote and a vertical asymptote. Any graph of a rational function can be obtained from the reciprocal function $f(x)=\dfrac{1}{x}$ by a combination of transformations, including a translation, stretches and compressions. Type 1: Vertical Compression \( y=\dfrac{a}{x}, \ 0 \lt a \lt 1 \) The […]
Translation Rule 1 For $y=f(x)+b$, the effect of $b$ is to translate the graph vertically through $b$ units. Translation Rule 2 For $y=f(x-a)$, the effect of $a$ is to translate the graph horizontally through $a$ units. Translation Rule 3 For $y=f(x-a)+b$, the graph is translated $a$ units horizontally and vertically $b$ units. Example 1 Given […]
These techniques involves sketching the graph of $y=\dfrac{1}{f(x)}$ from the graph of $y=f(x)$. Technique 1 When $f(x)$ approaches towards $0$, $y=\dfrac{1}{f(x)}$ approaches towards $\infty$, the graph of $y=\dfrac{1}{f(x)}$ approaches the vertical asymptote(s). Technique 2 The graph of $y=\dfrac{1}{f(x)}$ has vertical asymptotes at the $x$-intercepts of $y=f(x)$. Technique 3 When $f(x)$ approaches towards $\infty$, $y=\dfrac{1}{f(x)}$ approaches […]
Suppose we sketch the graphs of two functions $f(x)$ and $g(x)$ on the same axes. The $x$-coordinates of points where the graphs meet are the solutions to the equation $f(x)=g(x)$. We can use this property to solve equations graphically, but we must make sure the graphs are drawn carefully and accurately. Let’s take a look […]
Consider the graphs of a quadratic function and a linear function on the same set of axes. cutting2 points of intersection $b^2-4ac \gt 0$ touching1 point of intersection $b^2-4ac = 0$ missingno points of intersection $b^2-4ac \lt 0$ In the graphs meet, the coordinates of the points of intersection of the graphs can be found […]
Example 1 Find the equation of the quadratic graph. \( \begin{align} \displaystyley &= a(x+2)(x+1) \\2 &= a(0+2)(0+1) &\text{substitute }(0,2)\\2 &= 2a \\a &= 1 \\y &= 1(x+2)(x+1) \\\therefore y &= x^2+3x+2 \end{align} \) Example 2 Find the equation of the quadratic graph. \( \begin{align} \displaystyley &= a(x+1)^2 \\1 &= a(0+1)^2 &\text{substitute }(0,1)\\a &= 1 \\y […]
The equation of the axis of symmetry of $y=ax^2+bx+c$ is $x=-\dfrac{b}{2a}$. Example 1 Find the equation of the axis of symmetry of $y=x^2+4x-2$. \( \begin{align} \displaystylex &= -\dfrac{b}{2a} \\&= -\dfrac{4}{2 \times 1} \\\therefore x &= -2\end{align} \) Example 2 Find the equation of the axis of symmetry of $y=-2x^2-12x+3$. \( \begin{align} \displaystylex &= -\dfrac{b}{2a} \\&= […]
$x$-intercepts when $y=0$$y$-intercepts when $x=0$ Example 1 Sketch the graphs of $y=x^2+x-2$ by stating $x$- and $y$-intercepts. $x$-intercepts when $y=0$\( \begin{align} \displaystylex^2+x-2 &= 0 \\(x+2)(x-1) &= 0 \\x &= -2, 1 \\\end{align} \)$y$-intercepts when $x=0$\( \begin{align} \displaystyley &= 0^2+0-2 \\&= -2 \end{align} \) Example 2 Sketch the graphs of $y=-x^2+3$ by stating $x$- and $y$-intercepts. […]