Hyperbola

Curved Graphs

Hyperbola

Learning Objectives

• Plot the graph of hyperbola from a table of values / geogebra.org

• Interpret straight line and quadratic graphs and use them to solve related questions

Hyperbola

• Equations y=a/x are called reciprocal equations

• Graphs of reciprocal are called hyperbola

• Open your geogebra.org, draw y=2/x

Hyperbola y=2/x

• Has two equal parts, same size and shape

• Symmetrical

• Approach the axes, but never touche them

• No value of x when y=0, and for y when x=0 (no x-intecepts and no x-intercepts)

• A line that a graph approaches but never intersect is called asymptote

Hyperbola y=2/x

• Asymptote: x=0 and y=0

Hyperbola y=2/x-1

• Horizontal asymptote: y=0

• Vertical asymptote: x=1

• y-intercept = -2

Multiple Choice

Which of the followings is true?

• asymptote: y=0, x=1

• asymptote: y=0 x=-1

• asymptote: x=0, y=-1

• asymptote: x=-1, y=-1

Multiple Choice

Which of the followings is true?

• y-intercept =-1

• y-intercept = 1

• y-intercept = -3

• y-intercept =3

Multiple Choice

Which of the followings is true?

• asymptote: y=0, x=1

• asymptote: y=0 x=-1

• asymptote: x=0, y=2

• asymptote: x=3, y=0

Multiple Choice

Which of the followings is true?

• x-intercept =-1

• x-intercept = 1

• x-intercept = -3

• x-intercept =3

Using Graph to Solve Quadratics

• If the quadratic equation is in the general form it will equal zero: x²-2x-3=0 (draw y=x²-2x-3)​

• The solutions (roots) to the equation are the values of x where the curve crosses the x-axis as this is where y = 0. 

• solutions: x=-1 and x=3

The equations don't equal zero

• Not all quadratic equations equal zero.

• You can manipulate the equation to make it zero and then use the x-axis to find the solution or you can draw a second line on the graph.

• Drawing a second line, the intersection of two lines will be the solutions/roots

Example 1: x² = 5

• Rearrange the equation x² = 5 to equal zero by subtracting 5 from both sides: x² − 5 = 0

• The graph of y = x² − 5 is shown here.

• The solutions/roots = x-intercepts

• roots: x=-2.2 or x=2.2

Example 2: x² = 5

• Equation x² = 5, can be written as y = x² and y = 5

• Draw y = x² and y = 5

• The solutions/roots are the intersection of the two graphs, x=-2.2 and x =2.2

Using Known Graph

• If you have to solve an equation and you already have a graph, you can manipulate the equation so that you can use the graph to find the solution.

• Applying the previous method (drawing second line)

Example 3: Use graph of y=x²-3x-8 to solve x²-3x=12

• Manipulate: x²-3x-8=12-8

• We will have x²-3x-8=4 (y=x²-3x-8 and y=4)

• Draw the graphs, intersection are solutions/roots, x=-2.3 and x=5.3

Example 3: Use graph of y=x²-3x-8 to solve x²-3x=-3

• Manipulate: x²-3x-8=-3-8

• We will have x²-3x-8=-11 (y=x²-3x-8 and y=-11)

• Draw the graphs, intersection are solutions/roots, no intersection = no solutions/roots

Example 3: Use graph of y=x²-3x-8 to solve x²-3x-8=0

• Manipulate: x²-3x-8=0

• (y=x²-3x-8 and y=0)

• Draw the graphs, intersection are solutions/roots, no intersection = no solutions/roots

The number of Solutions/roots

• Curve crosses the axis:

  2 solutions

The number of Solutions/roots

• Curve touches the axis:

  1 solution

• Curve above the axis:

  no solutions