Exponential Functions and Graph Transformations with Constant Proportions

The lecture focuses on exponential functions, specifically y = e^(kx), and their graph transformations. It explains how multiplying the function or x-value affects the graph's stretch and shift. The gradient function is explored, showing that for y = e^(kx), the gradient is k times the function itself. The lecture generalizes this to any exponential function and discusses using exponential models when the gradient is proportional to the y-value.

What happens to the graph of a function when you multiply the x-value by a constant?

• It shifts left or right along the x-axis.

• It stretches along the y-axis by a scale factor of the constant.

• It stretches along the x-axis by a scale factor of 1 over the constant.

• It shifts up or down along the y-axis.

In the function y = e^(kx), what does the constant k represent in terms of graph transformation?

• The amount of vertical shift.

• The scale factor for stretching along the y-axis.

• The scale factor for stretching along the x-axis.

• The amount of horizontal shift.

How does the gradient of the graph y = e^(kx) relate to the function itself?

• The gradient is independent of the y-value.

• The gradient is the inverse of the y-value.

• The gradient is k times the y-value.

• The gradient is equal to the y-value.

What is the gradient function for y = e^(2x)?

• 2e^(2x)

• e^(2x)

• e^(x)

• 2x

When can an exponential model be used according to the lecture?

• When the rate of change is constant.

• When the rate of change is zero.

• When the rate of change is proportional to the y-value.

• When the rate of change is proportional to the x-value.

What is the relationship between the rate of change and the y-value in an exponential model?

• The rate of change is unrelated to the y-value.

• The rate of change is directly proportional to the y-value.

• The rate of change is equal to the y-value.

• The rate of change is inversely proportional to the y-value.

If y = e^(3x), what is the gradient of the graph?

• e^(x)

• 3x

• e^(3x)

• 3e^(3x)