Question

1. write an exponential equation in the form $y = ab^x$ whose graph passes through points $(0, 2)$ and $(1, 1.3)$

$y = 2(0.65)^x$
$y = 1(1.3)^x$
$y = (1.3)^x$
$y = 0.65(2)^x$

Explanation:

Answer:

To determine the exponential equation \( y = ab^x \) that passes through the points \( (0, 2) \) and \( (1, 1.3) \), we follow these steps:

Step 1: Use the point \( (0, 2) \) to find \( a \)

When \( x = 0 \), \( y = 2 \). Substituting into the equation \( y = ab^x \):

$$ 2 = ab^0 $$

Since \( b^0 = 1 \) for any \( b
eq 0 \), we get:

$$ a = 2 $$

Step 2: Use the point \( (1, 1.3) \) to find \( b \)

Now we know \( a = 2 \), and we use the point \( (1, 1.3) \) (where \( x = 1 \), \( y = 1.3 \)):

$$ 1.3 = 2 \cdot b^1 $$

Simplifying \( b^1 = b \), we solve for \( b \):

$$ b = \frac{1.3}{2} = 0.65 $$

Step 3: Write the exponential equation

Substituting \( a = 2 \) and \( b = 0.65 \) into \( y = ab^x \), we get:

$$ y = 2(0.65)^x $$

So the correct answer is the first option: \( \boldsymbol{y = 2(0.65)^x} \)