Question

identify the transformations made on f(x) = 10^x to create the graph of $-\frac{2}{3}(10)^{x - 8}$
options:

• reflection over y - axis, vertical compression by 2/3, right 8
• reflection over x - axis, vertical compression by 2/3, left 8
• reflection over x - axis, vertical compression by 2/3, right 8
• reflection over x - axis, vertical stretch by 3/2, right 8

Explanation:

Step1: Analyze reflection

The negative sign in front of the function \(-\frac{2}{3}(10)^{x - 8}\) indicates a reflection over the \(x\)-axis (since reflecting \(y = f(x)\) over \(x\)-axis gives \(y=-f(x)\)).

Step2: Analyze vertical compression/stretch

The coefficient \(\frac{2}{3}\) (with \(0<\frac{2}{3}<1\)) indicates a vertical compression by a factor of \(\frac{2}{3}\) (vertical compression when \(0 < a<1\) for \(y = a\cdot f(x)\)).

Step3: Analyze horizontal shift

The exponent is \(x - 8\), for a function \(y = f(x - h)\), it is a shift to the right by \(h\) units. So \(x-8\) means a shift right by 8 units.

Now let's check the options:

• First option: Reflection over \(y\)-axis (incorrect, we have reflection over \(x\)-axis), so eliminate.
• Second option: Shift left 8 (incorrect, we have shift right 8), eliminate.
• Third option: Reflection over \(x\)-axis, vertical compression by \(\frac{2}{3}\), right 8 (matches our analysis).
• Fourth option: Vertical stretch (incorrect, it's a compression) and factor wrong, eliminate.

Answer:

The correct option is the orange one: reflection over x - axis, vertical compression by 2/3, right 8