Answer:
C) The y-values increase by a factor of 7 for each x increase of 1.
Step-by-step explanation:
The table representing an exponential function is:
[tex]\begin{array}{|c|c|}\cline{1-2} x & y \\\cline{1-2} 0 & 1 \\\cline{1-2} 2 & 49 \\\cline{1-2} 4 & 2401 \\\cline{1-2} 6 & 117649 \\\cline{1-2} \end{array}[/tex]
The general form of a exponential function is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{General form of an Exponential Function}}\\\\\large\text{$f(x)=ab^x$}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ is the initial value ($y$-intercept).}\\ \phantom{ww}\bullet\;\textsf{$b$ is the base (growth/decay factor) in decimal form.}\end{array}}[/tex]
In this case, the value of a can be found by substituting point (0, 1) into the general equation:
[tex]1 = ab^0\\\\1=a(1)\\\\a=1[/tex]
To find the value of b, substitute another point from the table (2, 49) and a = 1 into the general equation:
[tex]49=(1)b^2\\\\b^2=49\\\\b=\sqrt{49}\\\\b=7[/tex]
So, the exponential equation that models the given data is:
[tex]\Large\boxed{\boxed{y = 7^x}}[/tex]
This means that each increase of 1 in the exponent (x-value) results in multiplying the previous value of y by 7:
[tex]x=1 \implies y=7^1 = 7\\\\x=2 \implies y=7^2 = 7 \times 7\\\\x=3 \implies y=7^3= 7 \times 7 \times 7\\\\x=4 \implies y=7^4= 7 \times 7 \times 7 \times 7[/tex]
Therefore, the correct statement describing how the y-values in the table grow is:
- The y-values increase by a factor of 7 for each x increase of 1.
