Graphing Exponential Functions: Understanding the Gradient Function

The video tutorial explores the gradient function of exponential graphs, focusing on y=a^x. It explains how the gradient increases with the value of 'a' and analyzes specific cases like y=2^x and y=3^x. The tutorial derives a general rule for calculating the gradient using a constant, and concludes with a derivation from first principles.

What happens to the steepness of the graph as the value of 'a' increases in y = a^x?

• The graph becomes a straight line.

• The graph becomes steeper.

• The graph remains unchanged.

• The graph becomes less steep.

Why do exponential graphs like y = a^x pass through the point (0, 1)?

• Because a^1 is always 0.

• Because a^0 is always 0.

• Because a^0 is always 1.

• Because a^1 is always 1.

In the function y = 2^x, what happens to the gradient as x increases?

• The gradient decreases.

• The gradient remains constant.

• The gradient increases.

• The gradient becomes zero.

What is the constant value that affects the gradient in y = 2^x?

• 1.2

• 1.0

• 0.7

• 0.5

For the function y = 3^x, what is the constant used to find the gradient?

• 0.9

• 1.0

• 1.1

• 1.2

How do you calculate the gradient for y = 3^x at a specific point?

• Multiply 3^x by 0.7

• Multiply 3^x by 2.0

• Multiply 3^x by 1.1

• Multiply 3^x by 1.5

What is the general formula for the gradient function of y = a^x?

• x times a^m

• a times m^x

• m times x^a

• m times a^x

In the derivation of the gradient function, what does H tend towards?

• Negative infinity

• One

• Zero

• Infinity

What is the significance of the constant 'm' in the gradient function?

• It is the slope of the tangent line.

• It is the y-intercept of the graph.

• It is the constant that multiplies the function.

• It is the x-intercept of the graph.

What is the relationship between 'm' and 'a' in the gradient function?

• m is the square of a.

• m is the cube of a.

• m is the logarithm of a.

• m is the reciprocal of a.