Question

4.
x | -1 | 0 | 1 | 2
y | 0.75 | 3 | 12 | 48

Explanation:

To determine the function that relates \( x \) and \( y \), we analyze the given table:

Step 1: Check for a pattern (exponential or linear)

Let's see the ratio between consecutive \( y \)-values:

• From \( x = -1 \) to \( x = 0 \): \( \frac{3}{0.75} = 4 \)
• From \( x = 0 \) to \( x = 1 \): \( \frac{12}{3} = 4 \)
• From \( x = 1 \) to \( x = 2 \): \( \frac{48}{12} = 4 \)

The common ratio is \( 4 \), so this is an exponential function.

Step 2: General form of an exponential function

The general form of an exponential function is \( y = ab^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base (common ratio).

Step 3: Determine \( a \) and \( b \)

• When \( x = 0 \), \( y = 3 \). So, \( a = 3 \) (since \( y = ab^0 = a \cdot 1 = a \)).
• The common ratio \( b = 4 \) (from the ratios calculated above).

Thus, the function is \( y = 3 \cdot 4^x \).

Let's verify with \( x = -1 \):
\( y = 3 \cdot 4^{-1} = 3 \cdot \frac{1}{4} = 0.75 \), which matches the table.

With \( x = 1 \):
\( y = 3 \cdot 4^1 = 12 \), which matches.

With \( x = 2 \):
\( y = 3 \cdot 4^2 = 3 \cdot 16 = 48 \), which also matches.

So the function is \( y = 3 \cdot 4^x \).

If the question was to find the equation of the function, the answer is \( \boldsymbol{y = 3 \cdot 4^x} \).

Answer:

To determine the function that relates \( x \) and \( y \), we analyze the given table:

Step 1: Check for a pattern (exponential or linear)

Let's see the ratio between consecutive \( y \)-values:

• From \( x = -1 \) to \( x = 0 \): \( \frac{3}{0.75} = 4 \)
• From \( x = 0 \) to \( x = 1 \): \( \frac{12}{3} = 4 \)
• From \( x = 1 \) to \( x = 2 \): \( \frac{48}{12} = 4 \)

The common ratio is \( 4 \), so this is an exponential function.

Step 2: General form of an exponential function

The general form of an exponential function is \( y = ab^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base (common ratio).

Step 3: Determine \( a \) and \( b \)

• When \( x = 0 \), \( y = 3 \). So, \( a = 3 \) (since \( y = ab^0 = a \cdot 1 = a \)).
• The common ratio \( b = 4 \) (from the ratios calculated above).

Thus, the function is \( y = 3 \cdot 4^x \).

Let's verify with \( x = -1 \):
\( y = 3 \cdot 4^{-1} = 3 \cdot \frac{1}{4} = 0.75 \), which matches the table.

With \( x = 1 \):
\( y = 3 \cdot 4^1 = 12 \), which matches.

With \( x = 2 \):
\( y = 3 \cdot 4^2 = 3 \cdot 16 = 48 \), which also matches.

So the function is \( y = 3 \cdot 4^x \).

If the question was to find the equation of the function, the answer is \( \boldsymbol{y = 3 \cdot 4^x} \).