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
You should know the following charecteristics of functions:
The 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.
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.
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.
Only for graphs of functions whose highest power is more than 1. There 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.
An asymptote is a straight or curved line, which the graph of a function will approach, but never touch.
A line about which the graph is symmetric
The interval on which a graph increases or decreases
A graph is said to be continuous if there are no breaks in the graph.
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’
You should know the following functions and their properties:
Functions of the form
$y=ax+q$ . These are straight lines.
Functions of the Form
$y=a{x}^{2}+q$ These are known as parabolic functions or parabolas.
Functions of the Form
$y=\frac{a}{x}+q$ . These are known as hyperbolic functions.
Functions of the Form
$y=a{b}^{\left(x\right)}+q$ . These are known as exponential functions.
End of chapter exercises
Sketch the following straight lines:
$y=2x+4$
$y-x=0$
$y=-\frac{1}{2}x+2$
Sketch the following functions:
$y={x}^{2}+3$
$y=\frac{1}{2}{x}^{2}+4$
$y=2{x}^{2}-4$
Sketch the following functions and identify the asymptotes:
$y={3}^{x}+2$
$y=-4.{2}^{x}+1$
$y=2.{3}^{x}-2$
Sketch the following functions and identify the asymptotes:
$y=\frac{3}{x}+4$
$y=\frac{1}{x}$
$y=\frac{2}{x}-2$
Determine whether the following statements are true or false. If the statement is false, give reasons why:
The given or chosen y-value is known as the independent variable.
An intercept is the point at which a graph intersects itself.
There are two types of turning points – minimal and maximal.
A graph is said to be congruent if there are no breaks in the graph.
Functions of the form
$y=ax+q$ are straight lines.
Functions of the form
$y=\frac{a}{x}+q$ are exponential functions.
An asymptote is a straight or curved line which a graph will intersect once.
Given a function of the form
$y=ax+q$ , to find the y-intersect put
$x=0$ and solve for
$y$ .
The graph of a straight line always has a turning point.
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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