Grade 10 Mathematics: Hyperbolas and Exponential Functions

The standard equation for a hyperbola in Grade 10 is defined as $y = \frac{a}{x} + q$ .
The graph consists of two separate branches that are located in diagonally opposite quadrants.
The position of these branches is determined by the sign of the parameter $a$ :
- If $a > 0$ , the branches of the hyperbola lie in the first and third quadrants.
- If $a < 0$ , the branches of the hyperbola lie in the second and fourth quadrants.
The magnitude of $a$ determines the distance of the curves from the intersection of the asymptotes. A larger absolute value of $a$ (represented as $|a|$ ) means the curves are further away from the origin.

An asymptote is a straight line that a curve approaches indefinitely but never actually touches or crosses.
In the function $y = \frac{a}{x} + q$ , there are two primary asymptotes:
- Vertical Asymptote: This occurs at $x = 0$ . Because division by zero is undefined in mathematics, the function has no value when the denominator is zero. Thus, the graph will never touch the y-axis.
- Horizontal Asymptote: This occurs at $y = q$ . As the value of $x$ becomes extremely large (positive or negative), the term $\frac{a}{x}$ approaches zero, meaning the y-value of the function approaches $q$ . The graph will never actually reach the value of $q$ .

The general equation for an exponential function is $y = a \cdot b^x + q$ .
Key constraints for the base $b$ are:
- $b > 0$ : The base must be positive.
- $b \neq 1$ : If the base were 1, the graph would simply be a horizontal line ( $1^x = 1$ ).
The shape of the graph depends on the value of $b$ :
- Exponential Growth: If $b > 1$ , the function values increase as $x$ increases.
- Exponential Decay: If $0 < b < 1$ , the function values decrease as $x$ increases.
The parameter $a$ affects the orientation and steepness:
- If $a > 0$ , the graph is above the horizontal asymptote.
- If $a < 0$ , the graph is reflected across the asymptote and lies below it.

The parameter $q$ in both hyperbolic and exponential functions represents a vertical shift.
Upward Shift: If $q > 0$ , the entire graph moves upward by a distance of $q$ units.
Downward Shift: If $q < 0$ , the entire graph moves downward by the absolute value of $q$ units.
In a hyperbola, shifting the graph vertically also moves the horizontal asymptote to the line $y = q$ .
In an exponential graph, the horizontal asymptote is defined by the equation $y = q$ . Changing $q$ moves the boundary line that the exponential curve approaches.

Hyperbola ( $y = \frac{a}{x} + q$ ):
- Domain: The set of all possible x-values. For a hyperbola, $x$ can be any real number except zero. This is written as $\{x \in \mathbb{R} : x \neq 0\}$ .
- Range: The set of all possible y-values. For a hyperbola, $y$ can be any real number except the value of the horizontal asymptote. This is written as $\{y \in \mathbb{R} : y \neq q\}$ .
Exponential Graph ( $y = a \cdot b^x + q$ where $a > 0$ ):
- Domain: The x-values can include any real number. This is written as $\{x \in \mathbb{R}\}$ .
- Range: Since the graph never touches the asymptote and (assuming $a > 0$ ) stays above it, the range is all values greater than $q$ . This is written as $\{y \in \mathbb{R} : y > q\}$ .

To determine the equation of an exponential function from a given graph, follow this step-by-step procedure:
1. Identify the Horizontal Asymptote: Look at the graph to see which horizontal line the curve is approaching. This value is $q$ . Substitute this into the equation $y = a \times b^x + q$ .
2. Identify the y-intercept: If the y-intercept is provided as a point $(0; y)$ , substitute these coordinates into the equation. Since any number (except zero) to the power of $0$ is $1$ ( $b^0 = 1$ ), you can solve for $a$ using the equation $y = a(1) + q$ .
3. Solve for the Base ( $b$ ): If $b$ is unknown but you have another point on the graph $(x; y)$ , substitute the calculated values of $a$ and $q$ along with the coordinates of the point into the equation. Use algebraic manipulation or logarithms to solve for $b$ .