QuestionIs the function $H(x)$ exponential? If yes, find the base $a$ . Given points: $(-1, 7)$ , $(0, 9)$ , $(1, 11)$ , $(2, 13)$ , $(3, 15)$ .

Studdy Solution

STEP 1

Assumptions1. The given function is represented as a table of values for $x$ and $H(x)$ . . An exponential function is of the form $H(x) = a \cdot b^x$ where $a$ and $b$ are constants, $a$ is the initial value, and $b$ is the base of the exponential function.
3. To determine if a function is exponential, we need to check if the ratio of consecutive $H(x)$ values is constant.

STEP 2

We start by calculating the ratio of consecutive $H(x)$ values. The ratio is given by $Ratio = \frac{H(x_{i+1})}{H(x_i)}$

STEP 3

Calculate the ratio for the first pair of $H(x)$ values.
$Ratio_{-1,0} = \frac{H(0)}{H(-1)} = \frac{9}{7}$

STEP 4

Calculate the ratio for the second pair of $H(x)$ values.
$Ratio_{0,1} = \frac{H(1)}{H(0)} = \frac{11}{9}$

STEP 5

Calculate the ratio for the third pair of $H(x)$ values.
$Ratio_{1,2} = \frac{H(2)}{H(1)} = \frac{13}{11}$

STEP 6

Calculate the ratio for the fourth pair of $H(x)$ values.
$Ratio_{2,3} = \frac{H(3)}{H(2)} = \frac{15}{13}$

STEP 7

Now, we compare the ratios. If the ratios are equal, the function is exponential, and the base $b$ is the value of the ratio. If the ratios are not equal, the function is not exponential.
$Ratio_{-1,0} = Ratio_{0,1} = Ratio_{1,2} = Ratio_{2,3}$

STEP 8

From the calculated ratios, we see that $\frac{}{7} \neq \frac{11}{} \neq \frac{13}{11} \neq \frac{15}{13}$ . Hence, the ratios are not equal.
Therefore, the given function is not an exponential function.

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QuestionIs the function $H(x)$ exponential? If yes, find the base $a$ . Given points: $(-1, 7)$ , $(0, 9)$ , $(1, 11)$ , $(2, 13)$ , $(3, 15)$ .

Studdy Solution

STEP 1

STEP 2

We start by calculating the ratio of consecutive $H(x)$ values. The ratio is given by $Ratio = \frac{H(x_{i+1})}{H(x_i)}$

STEP 3

Calculate the ratio for the first pair of $H(x)$ values.
$Ratio_{-1,0} = \frac{H(0)}{H(-1)} = \frac{9}{7}$

STEP 4

Calculate the ratio for the second pair of $H(x)$ values.
$Ratio_{0,1} = \frac{H(1)}{H(0)} = \frac{11}{9}$

STEP 5

Calculate the ratio for the third pair of $H(x)$ values.
$Ratio_{1,2} = \frac{H(2)}{H(1)} = \frac{13}{11}$

STEP 6

Calculate the ratio for the fourth pair of $H(x)$ values.
$Ratio_{2,3} = \frac{H(3)}{H(2)} = \frac{15}{13}$

STEP 7

Now, we compare the ratios. If the ratios are equal, the function is exponential, and the base $b$ is the value of the ratio. If the ratios are not equal, the function is not exponential.
$Ratio_{-1,0} = Ratio_{0,1} = Ratio_{1,2} = Ratio_{2,3}$

STEP 8

From the calculated ratios, we see that $\frac{}{7} \neq \frac{11}{} \neq \frac{13}{11} \neq \frac{15}{13}$ . Hence, the ratios are not equal.
Therefore, the given function is not an exponential function.

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QuestionIs the function H(x)H(x)H(x) exponential? If yes, find the base aaa. Given points: (−1,7)(-1, 7)(−1,7), (0,9)(0, 9)(0,9), (1,11)(1, 11)(1,11), (2,13)(2, 13)(2,13), (3,15)(3, 15)(3,15).

Studdy Solution

STEP 1

STEP 2

We start by calculating the ratio of consecutive H(x)H(x)H(x) values. The ratio is given byRatio=H(xi+1)H(xi)Ratio = \frac{H(x_{i+1})}{H(x_i)}Ratio=H(xi​)H(xi+1​)​

STEP 3

Calculate the ratio for the first pair of H(x)H(x)H(x) values.Ratio−1,0=H(0)H(−1)=97Ratio_{-1,0} = \frac{H(0)}{H(-1)} = \frac{9}{7}Ratio−1,0​=H(−1)H(0)​=79​

STEP 4

Calculate the ratio for the second pair of H(x)H(x)H(x) values.Ratio0,1=H(1)H(0)=119Ratio_{0,1} = \frac{H(1)}{H(0)} = \frac{11}{9}Ratio0,1​=H(0)H(1)​=911​

STEP 5

Calculate the ratio for the third pair of H(x)H(x)H(x) values.Ratio1,2=H(2)H(1)=1311Ratio_{1,2} = \frac{H(2)}{H(1)} = \frac{13}{11}Ratio1,2​=H(1)H(2)​=1113​

STEP 6

Calculate the ratio for the fourth pair of H(x)H(x)H(x) values.Ratio2,3=H(3)H(2)=1513Ratio_{2,3} = \frac{H(3)}{H(2)} = \frac{15}{13}Ratio2,3​=H(2)H(3)​=1315​

STEP 7

STEP 8

From the calculated ratios, we see that 7≠11≠1311≠1513\frac{}{7} \neq \frac{11}{} \neq \frac{13}{11} \neq \frac{15}{13}7​=11​=1113​=1315​. Hence, the ratios are not equal.Therefore, the given function is not an exponential function.

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QuestionIs the function $H(x)$ exponential? If yes, find the base $a$ . Given points: $(-1, 7)$ , $(0, 9)$ , $(1, 11)$ , $(2, 13)$ , $(3, 15)$ .

We start by calculating the ratio of consecutive $H(x)$ values. The ratio is given by $Ratio = \frac{H(x_{i+1})}{H(x_i)}$

Calculate the ratio for the first pair of $H(x)$ values.
$Ratio_{-1,0} = \frac{H(0)}{H(-1)} = \frac{9}{7}$

Calculate the ratio for the second pair of $H(x)$ values.
$Ratio_{0,1} = \frac{H(1)}{H(0)} = \frac{11}{9}$

Calculate the ratio for the third pair of $H(x)$ values.
$Ratio_{1,2} = \frac{H(2)}{H(1)} = \frac{13}{11}$

Calculate the ratio for the fourth pair of $H(x)$ values.
$Ratio_{2,3} = \frac{H(3)}{H(2)} = \frac{15}{13}$

From the calculated ratios, we see that $\frac{}{7} \neq \frac{11}{} \neq \frac{13}{11} \neq \frac{15}{13}$ . Hence, the ratios are not equal.
Therefore, the given function is not an exponential function.