Types of Graphs

Why do we need to know what graphs look like?

• 图中使用various aspects of mathematics – but in the real world they can take on specific meanings
• For example alinear (straight line)graph could be the path a ship needs to sail along to get from one port to another
• Anexponentialgraph (y=kx{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals k to the power of x" style="vertical-align:-6px;height:23px;width:46px" loading="lazy">) can be used to model population growth – for instance to monitor wildlife conservation projects

What are the shapes of graphs that we need to know?

• Recalling facts alone won’t do much for boosting your GCSE Mathematics grade!
• But being familiar with the general shapes of graphs will help you quickly recognise the sort of maths you are dealing with and features of the graph a question may refer to
• Below the basic form of the five types of function (other thantrig graphs) you need to recognise;
  • linear(y=±x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals plus-or-minus x" style="vertical-align:-6px;height:22px;width:54px" loading="lazy">)
  • quadratic(y=±x2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals plus-or-minus x squared" style="vertical-align:-6px;height:23px;width:61px" loading="lazy">)
  • cubic(y=±x3{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals plus-or-minus x cubed" style="vertical-align:-6px;height:23px;width:61px" loading="lazy">)
  • reciprocal(y=±1x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals plus-or-minus 1 over x" style="vertical-align:-17px;height:47px;width:62px" loading="lazy">)
  • exponential(y=k±x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals k to the power of plus-or-minus x end exponent" style="vertical-align:-6px;height:23px;width:55px" loading="lazy">)

• In addition, you need to recognise the three basic trigonometric graphs- but these are dealt with in the next section

A quadratic is a function of the formy=ax2+bx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals a x squared plus b x plus c" style="vertical-align:-6px;height:23px;width:115px" loading="lazy">wherea{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">is not zero
They are a very common type of function in mathematics, so it is important to know their key features

What does a quadratic graph look like?

• The shape made by a quadratic graph is known as aparabola
• The parabola shape of a quadratic graph can either look like a “u-shape” or an “n-shape”
  
  • A quadratic with apositive coefficientofx2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x squared" style="vertical-align:-6px;height:23px;width:18px" loading="lazy">will be au-shape
  • A quadratic with anegative coefficientofx2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x squared" style="vertical-align:-6px;height:23px;width:18px" loading="lazy">will be ann-shape
• A quadratic will always cross they{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis
• A quadratic may cross thex{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis twice, once, or not at all
  • The points where the graph crosses thex{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-axis are called theroots
• If the quadratic is au-shape, it has aminimum point(the bottom of the u)
• If the quadratic is ann-shape, it has amaximum point(the top of the n)
• Minimum and maximum points are both examples ofturning points

How do I sketch a quadratic graph?

• We could create a table of values for the function and then plot it accurately, however we often only require a sketch to be drawn, showing just the key features
• The most important features of a quadratic are
  • Its overall shape; a u-shape or an n-shape
  • Itsy{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-intercept
  • Itsx{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-intercept(s), these are also known as the roots
  • Its minimum or maximum point (turning point)
• If it is a positive quadratic (a{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">inax2+bx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a x squared plus b x plus c" style="vertical-align:-6px;height:23px;width:88px" loading="lazy">is positive) it will be a u-shape
• If it is a negative quadratic (a{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">inax2+bx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a x squared plus b x plus c" style="vertical-align:-6px;height:23px;width:88px" loading="lazy">is negative) it will be an n-shape
• They{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-intercept ofy=ax2+bx+c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals a x squared plus b x plus c" style="vertical-align:-6px;height:23px;width:115px" loading="lazy">will be0, c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 0 comma space c close parentheses" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">
• The roots, or thex{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">-intercepts will be the solutions toy=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals 0" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">;ax2+bx+c=0{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a x squared plus b x plus c equals 0" style="vertical-align:-6px;height:23px;width:114px" loading="lazy">
  • You can solve a quadratic by factorising, completing the square, or using the quadratic formula
  • There may be 2, 1, or 0 solutions and therefore 2, 1, or 0 roots
• The minimum or maximum point of a quadratic can be found by;
  • Completing the square
    • Once the quadratic has been written in the formy=px-q2+r{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals p open parentheses x minus q close parentheses squared plus r" style="vertical-align:-6px;height:23px;width:118px" loading="lazy">, the minimum or maximum point is given byq, r{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses q comma space r close parentheses" style="vertical-align:-6px;height:22px;width:39px" loading="lazy">
    • Be careful with the sign of thex-coordinate. E.g. if the equation isy=x-32+2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals open parentheses x minus 3 close parentheses squared plus 2" style="vertical-align:-6px;height:23px;width:109px" loading="lazy">then the minimum point is3, 2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="open parentheses 3 comma space 2 close parentheses" style="vertical-align:-6px;height:22px;width:40px" loading="lazy">but if the equation isy=x+32+2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals open parentheses x plus 3 close parentheses squared plus 2" style="vertical-align:-6px;height:23px;width:109px" loading="lazy">then the minimum point is-3, 2{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="开括号- 3逗号space 2 close parentheses" style="vertical-align:-6px;height:22px;width:53px" loading="lazy">

How do we draw a graph using a table of values in a non-calculator exam?

• Before you start, think what the graph might look like- see the previous notes on being familiar with shapes of graphs
• Using the rules of BIDMAS/ order of operations, substitute eachx{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">- value into the given function
• PLOT POINTS and join with a SMOOTH CURVE
• If there are any points that don't seem to fit with the shape of the rest of the curve, check your calculations for them again!

How do we draw a graph using a table of values in a calculator exam?

• Before you start, think what the graph might look like – see the previous notes on being familiar with shapes of graphs
• Find theTABLEfunction on your CALCULATOR
• Enter the FUNCTION – f(x)
  (use ALPHA button andxorX, depending on make/model)

  (Press = when finished)

  (If you are asked for another function, g(x), just press enter again)
• EnterStart,EndandStep(gap betweenxvalues)
• Press = and scroll up and down to seeyvalues
• PLOT POINTS and join with a SMOOTH CURVE
• To avoid errors always put negative numbers in brackets and use the (-) key rather than the subtraction key
• If your calculator does not have a TABLE function, then you will have to work out eachyvalue separately using the normal mode on your calculator