Graphs & Transformations

Part of the Oxford MAT Livestream

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MAT syllabus

The graphs of quadratics and cubics. Graphs of
\begin{equation*}
\sin x, \quad \cos x, \quad \tan x, \quad \sqrt{x}, \quad a^x ,\quad \log_a x.
\end{equation*}
Solving equations and inequalities with graphs.

The relations between the graphs
\begin{equation*}
y = f (ax),\quad y = af (x),\quad y = f (x - a), \quad y = f (x) + a
\end{equation*}
and the graph of $y = f (x)$.

Revision

  • The graph of an equation involving $x$ and $y$ is all the points in the $(x,y)$ plane that satisfy the equation. For a function $f(x)$, the graph of $y=f(x)$ shows the value of $f$ at each value of $x$.
  • Quadratics $y=ax^2+bx+c$ have graphs like these

Two parabolas  - the left one curves like a horse-shoe pointing up, the right one points down

 

  • Cubics can have 0 or 1 or 2 turning points.

Three cubics. The first increases from left to right, curving slightly to be a little flatter in the middle. The second decreases and has a point in the middle where it's flat. The third has two turning points; it increases then decreases then increases.

  • Other polynomials have graphs that might have more turning points (up to $(n-1)$ turning points if $x^n$ is the highest power of $x$ in the polynomial)
  • Graphs of $y=\sin x$ (red solid line) and $y=\cos x$ (green dashed line) and $y=\tan x$ (blue dot-dashed line);

A red curve rises then falls then rises. A green dashed curve falls then rises. A blue dot-dashed line only increases, but it goes off the top of the image at 90 and at 270, starting again at the bottom of the image each time.

  • Here are some more graphs. Note that $\sqrt{x}=x^{1/2}$ so the derivative is $\frac{1}{2}x^{-1/2}$, which is very large near $x=0$.
$y=\sqrt{x}$ $y=a^x$ with $a>1$ $y=a^{-x}$ with $0<a<1$
A curve that initially increases quickly near x=0, but the rate of increase slows down. A curve that increases faster and faster for larger x A curve that decreases, but the rate of decrease is slower and slower for larger x

 

  • Here's the graph of $\log_a x$. Note that $\log_a x$ is very negative for $x$ close to zero.

A curve that is very steep near x=0, and the value is very negative there, before the curve rises through (1,0) and increases further, but at a slower rate for larger x.

  • The graph of $y=f(x-a)$ is the translation of the graph of $y=f(x)$ by a distance $a$ in the positive $x$-direction.
  • The graph of $y=f(x)+a$ is the translation of the graph of $y=f(x)$ by a distance $a$ in the positive $y$-direction.
  • The graph of $y=f(ax)$ is a stretch of the graph of $y=f(x)$ by a factor of $\frac{1}{a}$ parallel to the $x$-axis.
  • The graph of $y=af(x)$ is a stretch of the graph of $y=f(x)$ by a factor of $a$ parallel to the $y$-axis.


Warm-up

  • Sketch $y=ax^2+bx+c$ in the following eight cases;
    • $a>0$, $b>0$, $c>0$
    • $a>0$, $b>0$, $c<0$
    • $a>0$, $b<0$, $c>0$
    • $a>0$, $b<0$, $c<0$
    • $a<0$, $b>0$, $c>0$
    • $a<0$, $b>0$, $c<0$
    • $a<0$, $b<0$, $c>0$
    • $a<0$, $b<0$, $c<0$
      In each case, decide how many roots there could be (there might be more than one possible sketch to draw).
  • Let $f(x)=x^2+4x+3$. Sketch the graph of $y=f(x+2)$.
  • Sketch the graph of $y=3 f( 2 x)$.
  • Sketch the graph of $y=2 f( 3 x)$. Is that the same as the previous graph?
  • Let $f(x)=x^3-x$. Sketch the graph of $y=2f(x+1)$.
  • Sketch the graph of $y=2f(x)+1$. Is that the same as the previous graph?
  • Sketch the graph of $y=\log_2 x$. Sketch a graph of $y=\log_2 (x^2)$.
  • Sketch the graph of $y=2^x$. Sketch a graph of $y=\left(\frac{1}{3}\right)^{x}$.
  • Sketch the graph of $y= x\sqrt{2}$. Does the graph go through the point $\displaystyle \left(\frac{p}{q},\frac{r}{s}\right)$ for any positive whole numbers $p$, $q$, $r$, $s$?
  • Sketch the graph of $y=\cos 2x$. Sketch a graph of $y=\frac{1}{2} + \frac{1}{2}\cos 2x$.
  • Sketch all the points $(x,y)$ that satisfy $x+y=3$.
  • Sketch all the points $(x,y)$ that satisfy $y^2=9x$.
  • Sketch all the points $(x,y)$ that satisfy $x^2=4-y^2$.


MAT questions

MAT 2015 Q1I

Into how many regions is the plane divided when the following equations are graphed, not considering the axes?
\begin{eqnarray*}
y = x^3\\
y = x^4\\
y = x^5
\end{eqnarray*}

(a) 6,

(b) 7,

(c) 8,

(d) 9,

(e) 10.


Hint: Find any points of intersection. You might find that there are one or two points that lie on all three graphs. Don't forget to count any unbounded regions.

 

MAT 2014 Q1B

The graph of the function $y=\log_{10}\left(x^2-2x+2\right)$ is sketched in

(a) A curve dips down then up again, with minimum at about x=1 and y somewhere between 0.2 and 0.5

(b) A curve dips down and then up again, with minimum at x=-1, y=0

(c) A curve that is not defined for x=1. After x=1 it increases like a logarithm, and before x=1 it decreases like log(-x)

(d) A curve that is not defined for x=-1. It increases like a logarithm from x=-1, and for x less than minus 1 it decreases like log(-x)

(e) A curve dips down then rises, with minimum at x=1, y=0

 

Hint: It would be good to know if the function inside the logarithm is ever zero.

 

MAT 2017 Q1D

The diagram below shows the graph of $y = f(x)$.

A curve that is not defined at x=1, and which looks like y=-x for large x and for very negative x

The graph of the function $y = -f(-x)$ is drawn in which of the following diagrams?

(a) A curve which is not defined for x=-1 and which looks like y=x for large x and for very negative x

(b) A curve which is not defined for x=1 and which looks like y=x for large x and for very negative x

(c) A curve which is not defined for x=-1 and which looks like y=-x for large x and for very negative x

(d) A curve which is not defined for x=1 and which looks like y=x for large x and looks like y=-x for very negative x

(e) A curve which is not defined for x=-1 and which looks like y=x for large x and like y=-x for very negative x

Hint: try to get to $-f(-x)$ in two steps, drawing pictures of what it looks like along the way.


MAT 2013 Q3

Let $0<k<2$. Below is sketched a graph of $y=f(x)$ where $f_k(x)=x(x-k)(x-2)$. Let $A(k)$ denote the area of the shaded region.

A cubic with roots at 0 and k and 2. The region between the curve and the x axis (in two parts - below and above k) is shaded grey.

(i) Without evaluating them, write down an expression for $A(k)$ in terms of two integrals.

(ii) Explain why $A(k)$ is a polynomial in $k$ of degree 4 or less. You are not required to calculate $A(k)$ explicitly.

(iii) Verify that $f_k(1+t)=-f_{2-k}(1-t)$ for any $t$.

(iv) How can the graph of $y=f_k(x)$ be transformed to the graph of $y=f_{2-k}(x)$? Deduce that $A(k)=A(2-k)$.

(v) Explain why there are constants $a$, $b$, $c$ such that \[A(k)=a(k-1)^4+b(k-1)^2+c.\] You are not required to calculate $a$, $b$, $c$ explicitly.

Hints: The first two parts aren't about graphs or transformations, so you could skip them if you're not interested in practising integration right now. This happens quite often, that the later parts of a MAT question can be approached even if you haven't done the previous parts.

In part (iii), note that we prove this with algebra (we know what $f_k(x)$ is, so just plug in the values). We'll need to be careful with the right-hand-side; $f_{2-k}(x)$ involves replacing each $k$ in the definition of $f_k$ with $(2-k)$. In general, it's important to know that you can sometimes use geometry or arguments about the symmetry of a graph, and sometimes you can just calculate away and do things like this with algebra.

In part (iv), you might need to apply more than one transformation; transform and then transform again!

Extension

The following material is included for your interest only, and not for MAT preparation.

There are lots of other interesting functions or equations that we might want to sketch. Here are some from the Oxford Online Maths Club, in no particular order and without context.

$$y=e^{-x}\cos x +\frac{1}{6}$$

$$y=e^{-x}\sin(x)$$

$$y=\sin(e^{-x})$$

$$y=\frac{x}{\ln x}\quad\text{and}\quad y=\frac{\ln x}{x}$$

$$y=\sin(x)-\sin(2x)\quad\text{and}\quad y=\frac{1}{2}\sin(x)-\frac{1}{16}\sin(2x)$$

$$y=x^3-\frac{35}{6}x^2+10x-\frac{25}{6}$$

$$y=\frac{1}{1+x^2}\quad\text{and}\quad y=\frac{1}{1+x^3}\quad\text{and}\quad y=\frac{1}{1+x^4}$$

$$y=\left(\frac{a}{x}\right)^{12}-\left(\frac{a}{x}\right)^6\quad \text{with $a>0$}$$

$$y=(\sin x)^2\quad\text{and}\quad y=(\sin x)^4\quad\text{and}\quad y=(\sin x)^6$$

$$y=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$

$$x^3+y^3=1$$

$$y=x^{1/2}\sin\left(\frac{1}{x}\right)$$

$$y=e^{-x}\left(x^2-4x+2\right)$$