Draw the graphs of
$y={2}^{x}$ and
$y={\left(\frac{1}{2}\right)}^{x}$ on the same set of axes.
Is the
$x$ -axis and asymptote or and axis of symmetry to both graphs ? Explain your answer.
Which graph is represented by the equation
$y={2}^{-x}$ ? Explain your answer.
Solve the equation
${2}^{x}={\left(\frac{1}{2}\right)}^{x}$ graphically and check that your answer is correct by using substitution.
Predict how the graph
$y=2.{2}^{x}$ will compare to
$y={2}^{x}$ and then draw the graph of
$y=2.{2}^{x}$ on the same set of axes.
The curve of the exponential function
$f$ in the accompanying diagram cuts the y-axis at the point A(0; 1) and B(2; 4) is on
$f$ .
Determine the equation of the function
$f$ .
Determine the equation of
$h$ , the function of which the curve is the reflection of the curve of
$f$ in the
$x$ -axis.
Determine the range of
$h$ .
Summary
Charecteristics of functions
1. dependent and independent variables2. domain and range
3. intercepts with axes4. turning points
5. asymptotes6. lines of symmetry
7. intervals on which the function increases/decreases8. continuous nature of the function
Dependent and independent variablesThe given or chosen x-value is known as the independent variable, because its value can be chosen freely. The calculated y-value is known as the dependent variable, because its value depends on the chosen x-value.
Domain and range
The domain of a relation is the set of all the x values for which there exists at least one y value according to that relation. The range is the set of all the y values, which can be obtained using at least one x value.
Set notation
A set of certain x values has the following form: {x : conditions, more conditions}
Interval notation
Here we write an interval in the form ’lower bracket, lower number, comma, upper number, upper bracket’
Intercepts with the axes
The intercept is the point at which a graph intersects an axis. The x-intercepts are the points at which the graph cuts the x-axis and the y-intercepts are the points at which the graph cuts the y-axis.
Turning points
Only for graphs of functions whose highest power is more than 1There are two types of turning points: a minimal turning point and a maximal turning point. A minimal turning point is a point on the graph where the graph stops decreasing in value and starts increasing in value and a maximal turning point is a point on the graph where the graph stops increasing in value and starts decreasing.
Asymptote
An asymptote is a straight or curved line, which the graph of a function will approach, but never touch.
Lines of symmetry
A line about which the graph is symmetric
Decreasing/increasing intervals
The interval on which a graph increases or decreases
Discrete and continuous nature
A graph is said to be continuous if there are no breaks in the graph.
Functions of the form y = ax + q
Straight lineTable summarising a and q
domain and range
Functions of the Form y = ax^2 + q
ParabolaTable summarising a and q
Domain is fixed, range variesTurning point present
Axis of symmetry through turning point
Functions of the Form y = a/x+q
Hyperbolic functionsTable summarising a and q
Domain and rangeAsymptotes
Functions of the Form y = ab^(x) + q
Exponential functionsTable summarising a and q
Domain fixed, range variesAsymptote
End of chapter exercises
Given the functions
$f\left(x\right)=-2{x}^{2}-18$ and
$g\left(x\right)=-2x+6$
Draw
$f$ and
$g$ on the same set of axes.
Calculate the points of intersection of
$f$ and
$g$ .
Hence use your graphs and the points of intersection to solve for
$x$ when:
$f\left(x\right)>0$
$\frac{f\left(x\right)}{g\left(x\right)}\le 0$
Give the equation of the reflection of
$f$ in the
$x$ -axis.
After a ball is dropped, the rebound height of each bounce decreases. The equation
$y=5\xb7{(0,8)}^{x}$ shows the relationship between
$x$ , the number of bounces, and
$y$ , the height of the bounce, for a certain ball. What is the approximate height of the fifth bounce of this ball to the nearest tenth of a unit ?
Mark had 15 coins in five Rand and two Rand pieces. He had 3 more R2-coins than R5-coins. He wrote a system of equations to represent this situation, letting
$x$ represent the number of five rand coins and
$y$ represent the number of two rand coins. Then he solved the system by graphing.
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