# Graphs with Gradient Proportional to One of the Coordinates

## Differential Equations Representing Graphs with Gradient Proportional to One of the Coordinates

- Exponential graphs frequently appear in higher level mathematics, often in the context of differential equations where the rate of change is proportional to the current value - typically written as
**dy/dx = ky**, where**k**is a constant. - In the differential equation
**dy/dx = ky**,**y**denotes the dependent variable,**x**is the independent variable, and the constant**k**determines the graph’s shape. When**k > 0**, the exponential graph shows growth, while for**k < 0**, the graph shows decay. - One solution to the differential equation
**dy/dx = ky**is**y = Ce**, where^{kx}**C**is the value of**y**when**x = 0**. This represents an exponential function where the gradient at any point is proportional to its y-coordinate.

## Sketching Exponential Graphs

- For
**k > 0**, the graph of**y = Ce**rises upwards as^{kx}**x**increases, asymptotic to the x-axis (y=0) as**x -> -∞**, and tends towards**∞**as**x -> ∞**. The gradient at any point on the graph is proportional to its y-coordinate and gets steeper as**x**increases. - For
**k < 0**, the graph of**y = Ce**falls towards the x-axis as^{kx}**x**increases, starts at**y = C**when**x = 0**and tends towards**0**as**x -> ∞**. The gradient at any point is proportional to the y-coordinate and gets less steep as**x**grows.

## Solving Problems Involving Exponential Growth or Decay

- In solving problems, it’s crucial to identify the
**growth or decay factor**in the situation at hand, and link this to the constant**k**. - Common examples might consider the population of bacteria growing exponentially, with
**k**indicating the growth rate, or radioactive decay, where**k**is the rate of decay. In either case, identifying**k**is key. - To find the value of
**C**, we often use an initial condition that provides the value of**y**at a specific**x**. From the general solution**y = Ce**, we substitute the given values for^{kx}**x**and**y**to find**C**.

## Using Logarithms to Determine Constants

- When provided with an exponential equation and an additional condition, and tasked with determining the constants, it can be beneficial to use the
**logarithm properties**to transform the exponential equation. This makes it simpler to solve for the desired constants. - For instance, to solve for
**k**in the equation**y = Ce**, after determining^{kx}**C**using an initial condition, the equation could be rewritten in logarithmic form using the equation**ln(y) = kx + ln(C)**. By substituting a second known**x**and**y**condition into this equation,**k**can be found.