To solve problems 4, 5, and 6, let us derive the equations for each exponential function:

Problem 4

From the graph:

• The function appears to pass through points $(-2, 3)$, $(0, 6)$, and $(2, 12)$.
• The general form of an exponential equation is $f(x) = a \cdot b^x$.

Step 1: Solve for $a$ using the point $(0, 6)$:

When $x = 0$, $f(x) = a \cdot b^0 = a \cdot 1 = 6$.
Thus, $a = 6$.

Step 2: Solve for $b$ using another point, e.g., $(2, 12)$:

Substitute $a = 6$ and $(x, f(x)) = (2, 12)$ into the equation:
$12 = 6 \cdot b^2.$
Divide by 6:
$b^2 = 2.$
Take the square root:
$b = \sqrt{2}.$

Final Equation:

$f(x) = 6 \cdot (\sqrt{2})^x.$

Problem 5

From the table:

• $x = 0, y = 5$,
• $x = 1, y = 10$,
• $x = 2, y = 20$,
• $x = 3, y = 40$.

Step 1: Use the general form $f(x) = a \cdot b^x$.

Using the point $(0, 5)$:
$f(0) = a \cdot b^0 = a = 5.$
Thus, $a = 5$.

Step 2: Solve for $b$ using another point, e.g., $(1, 10)$:

$10 = 5 \cdot b^1.$
Divide by 5:
$b = 2.$

Final Equation:

$f(x) = 5 \cdot 2^x.$

Problem 6

From the graph:

• The curve passes through points $(-2, 8)$, $(0, 4)$, and $(2, 2)$.
• The graph is decreasing, so $b < 1$.

Step 1: Use the general form $f(x) = a \cdot b^x$.

Using the point $(0, 4)$:
$f(0) = a \cdot b^0 = a = 4.$
Thus, $a = 4$.

Step 2: Solve for $b$ using another point, e.g., $(2, 2)$:

$2 = 4 \cdot b^2.$
Divide by 4:
$b^2 = \frac{1}{2}.$
Take the square root:
$b = \frac{1}{\sqrt{2}}.$

Final Equation:

$f(x) = 4 \cdot \left(\frac{1}{\sqrt{2}}\right)^x.$

Summary:

1. Problem 4: $f(x) = 6 \cdot (\sqrt{2})^x$,
2. Problem 5: $f(x) = 5 \cdot 2^x$,
3. Problem 6: $f(x) = 4 \cdot \left(\frac{1}{\sqrt{2}}\right)^x$.

Would you like a deeper explanation of any step?

Relative Questions:

1. How can you confirm the values of $b$ using additional points?
2. Why does $b < 1$ indicate exponential decay in Problem 6?
3. How would these equations change if $b$ were negative?
4. How does the asymptote affect the exponential graph's equation?
5. Could you use logarithms to find $b$ more efficiently in these problems?

Tip:

Always check multiple points from the graph or table to verify your exponential model.