Let’s solve the questions step by step:

Question 5

Estimate the gradient of the graph when $x = 3$:

1. To estimate the gradient, draw a tangent to the curve at $x = 3$. The tangent should just touch the curve at $x = 3$.
2. Identify two points on the tangent line and calculate the gradient using the formula: $\text{Gradient} = \frac{\Delta y}{\Delta x}$ From the graph, choose two points on the tangent. For instance:
   • Approximate coordinates: $(2, 2)$ and $(4, 4)$.
3. Substitute into the gradient formula: $\text{Gradient} = \frac{4 - 2}{4 - 2} = \frac{2}{2} = 1$

Thus, the estimated gradient at $x = 3$ is approximately 1.

Question 6

Find $a$ and $b$ given the equation $y = ab^x$ and points $(0, 5)$ and $(2, 45)$:

The general equation is $y = ab^x$. Use the points provided to set up two equations.

1. Point $(0, 5)$: Substituting into $y = ab^x$: $5 = ab^0$ Since $b^0 = 1$, we get: $a = 5$

2. Point $(2, 45)$: Substituting $a = 5$ into $y = ab^x$: $45 = 5b^2$ Simplify: $b^2 = \frac{45}{5} = 9$ Take the square root: $b = 3 \quad (\text{since } b > 0)$

Thus: $a = 5, \quad b = 3$

Final Answers:

• Question 5: Gradient at $x = 3$: 1
• Question 6: $a = 5, \, b = 3$

Would you like a detailed explanation or graph illustration for either solution? Here are related questions:

1. How do you accurately draw tangents to a curve?
2. How do logarithmic equations relate to solving exponential problems?
3. What are the implications of gradients in real-world applications?
4. How can you confirm your $a$ and $b$ values using substitution?
5. What role does the base $b > 0$ condition play in exponential functions?

Tip: For better accuracy, always use a ruler and check the units carefully when estimating gradients from a graph.