Formule bij exponentiële groei opstellen

\(y = b ⋅ g^{x}\) met \(g_{\text{week}} = 1 + {4{,}8 \over 100} = 1{,}048 \text{.}\)

De beginwaarde is de hoeveelheid bij \(x = 0 \text{,}\) dus \(b = 444 \text{.}\)

\(y = 444 ⋅ 1{,}048^{x} \text{.}\)

\(y = b ⋅ g^{x}\) met \(g = ({296 \over 327})^{{1 \over 4 - 2}} = 0{,}951...\)

\(\begin{rcases}y = b ⋅ 0{,}951...^{x} \\ x = 2 \text{ en } y = 327\end{rcases} \begin{matrix}b ⋅ 0{,}951...^{2} = 327 \\ b = {327 \over 0{,}951...^{2}} ≈ 361\end{matrix}\)

\(y = 361 ⋅ 0{,}951^{x} \text{.}\)

\(y = b ⋅ g^{x}\) met \(g = ({500 \over 438})^{{1 \over 6 - 2}} = 1{,}033...\)

\(\begin{rcases}y = b ⋅ 1{,}033...^{x} \\ x = 2 \text{ en } y = 438\end{rcases} \begin{matrix}b ⋅ 1{,}033...^{2} = 438 \\ b = {438 \over 1{,}033...^{2}} ≈ 410\end{matrix}\)

\(y = 410 ⋅ 1{,}034^{x} \text{.}\)

Rasterpunten \((2 , 90\,000)\) en \((7 , 400)\) aflezen.

\(y = b ⋅ g^{x}\) met \(g = ({400 \over 90\,000})^{{1 \over 7 - 2}} = 0{,}338...\)

\(\begin{rcases}y = b ⋅ 0{,}338...^{x} \\ x = 2 \text{ en } y = 90\,000{,}00\end{rcases} \begin{matrix}b ⋅ 0{,}338...^{2} = 90\,000{,}00 \\ b = {90\,000{,}00 \over 0{,}338...^{2}} \\ b = 785444{,}524...\end{matrix}\)

\(y = 785\,445 ⋅ 0{,}339^{x} \text{.}\)

\({22{,}82 \over 26{,}54} ≈ 0{,}86\)

\({19{,}63 \over 22{,}82} ≈ 0{,}86\)
\({16{,}88 \over 19{,}63} ≈ 0{,}86\)

De quotiënten zijn bij benadering gelijk, dus de tabel hoort bij een exponentieel verband.

\(y = b ⋅ g^{x}\) met \(g = 0{,}86\)

\(b\) is de waarde bij \(x = 0 \text{,}\) dus \(b = 26{,}54 \text{.}\)

Dus \(y = 26{,}54 ⋅ 0{,}86^{x} \text{.}\)

\(g = ({63{,}60 \over 103{,}56})^{{1 \over 7 - 4}} ≈ 0{,}85\)

\(g = ({23{,}99 \over 63{,}60})^{{1 \over 13 - 7}} ≈ 0{,}85\)
\(g = ({17{,}33 \over 23{,}99})^{{1 \over 15 - 13}} ≈ 0{,}85\)
\(g = ({14{,}73 \over 17{,}33})^{{1 \over 16 - 15}} ≈ 0{,}85\)

De groeifactoren zijn bij benadering gelijk, dus de tabel hoort bij een exponentieel verband.

\(y = b ⋅ g^{x}\) met \(g = 0{,}85\)

\(\begin{rcases}y = b ⋅ 0{,}85^{x} \\ x = 4 \text{ en } y = 103{,}56\end{rcases} \begin{matrix}b ⋅ 0{,}85^{4} = 103{,}56 \\ b = {103{,}56 \over 0{,}85^{4}} \\ b ≈ 198{,}39\end{matrix}\)

Dus \(y = 198{,}39 ⋅ 0{,}85^{x} \text{.}\)

\(y = a x + b\) met \(a = {\Delta y \over \Delta x} = {1 - 11 \over 12 - 11} = -10\)

\(\begin{rcases}y = -10 x + b \\ \text{door } (11 , 11)\end{rcases} \begin{matrix}-10 ⋅ 11 + b = 11 \\ -110 + b = 11 \\ b = 121\end{matrix}\)

Dus \(l{:}\,y = -10 x + 121\)

\(y = b ⋅ g^{x}\) met \(g = ({1 \over 11})^{{1 \over 12 - 11}} = 0{,}090...\)

\(\begin{rcases}y = b ⋅ 0{,}090...^{x} \\ \text{door } (11 , 11)\end{rcases} \begin{matrix}11 = b ⋅ 0{,}090...^{11} \\ b = {11 \over 0{,}090...^{11}} \\ b = 3\,138\,428\,376\,721\end{matrix}\)

Dus \(y = 3\,138\,428\,376\,721 ⋅ 0{,}091^{x} \text{.}\)

\({19{,}40 \over 19{,}60} ≈ 0{,}99\)

\({19{,}21 \over 19{,}40} ≈ 0{,}99\)
\({19{,}02 \over 19{,}21} ≈ 0{,}99\)

De quotiënten zijn bij benadering gelijk, dus de tabel hoort bij een exponentieel verband.

\(y = b ⋅ g^{x}\) met \(g = 0{,}99\)

\(b\) is de waarde bij \(x = 0 \text{,}\) dus \(b = 19{,}60 \text{.}\)

Dus \(y = 19{,}60 ⋅ 0{,}99^{x} \text{.}\)

\(g = ({83{,}68 \over 401{,}48})^{{1 \over 2\,017 - 2\,011}} ≈ 0{,}77\)

\(g = ({29{,}42 \over 83{,}68})^{{1 \over 2\,021 - 2\,017}} ≈ 0{,}77\)
\(g = ({17{,}44 \over 29{,}42})^{{1 \over 2\,023 - 2\,021}} ≈ 0{,}77\)

De groeifactoren zijn bij benadering gelijk, dus de tabel hoort bij een exponentieel verband.

\(y = b ⋅ g^{x}\) met \(g = 0{,}77\)

\(\begin{rcases}y = b ⋅ 0{,}77^{x} \\ x = 1 \text{ en } y = 401{,}48\end{rcases} \begin{matrix}b ⋅ 0{,}77^{1} = 401{,}48 \\ b = {401{,}48 \over 0{,}77^{1}} \\ b ≈ 521{,}40\end{matrix}\)

Dus \(y = 521{,}40 ⋅ 0{,}77^{x} \text{.}\)

Punt \(\text{A} (1 , 10\,000) \text{.}\)

Punt \(\text{B} (5 , 3\,000) \text{.}\)

Punt \(\text{C} (9 , 700) \text{.}\)

Rasterpunten \((2 , 3)\) en \((16 , 4)\) aflezen.

\(\log(y) = a x + b\) met \(a = {\Delta \log(y) \over \Delta x} = {4 - 3 \over 16 - 2} = \frac{1}{14}\)

\(\begin{rcases}\log(y) = \frac{1}{14} x + b \\ \text{door } (2 , 3)\end{rcases} \begin{matrix}\frac{1}{14} ⋅ 2 + b = 3 \\ \frac{2}{14} + b = 3 \\ b = 2\frac{6}{7}\end{matrix}\)

\(\log(y) = \frac{1}{14} x + 2\frac{6}{7}\)
\(y = 10^{\frac{1}{14} x + 2\frac{6}{7}}\)

\(y = 10^{\frac{1}{14} x} ⋅ 10^{2\frac{6}{7}}\)
\(\text{ } = 10^{2\frac{6}{7}} ⋅ (10^{\frac{1}{14}})^{x}\)
\(\text{ } = 720 ⋅ 1{,}179^{x}\)

Rasterpunten \((1 , 2)\) en \((4 , 1)\) aflezen.

\(\log(y) = a ⋅ \log(x) + b\) met \(a = {\Delta \log(y) \over \Delta \log(x)} = {1 - 2 \over 4 - 1} = -\frac{1}{3}\)

\(\begin{rcases}\log(y) = -\frac{1}{3} ⋅ \log(x) + b \\ \text{door } (1 , 2)\end{rcases} \begin{matrix}-\frac{1}{3} ⋅ 1 + b = 2 \\ -\frac{1}{3} + b = 2 \\ b = 2\frac{1}{3}\end{matrix}\)

\(\log(y) = -\frac{1}{3} ⋅ \log(x) + 2\frac{1}{3}\)
\(\log(y) = \log(x^{-\frac{1}{3}}) + \log(10^{2\frac{1}{3}})\)
\(\log(y) = \log(10^{2\frac{1}{3}} ⋅ x^{-\frac{1}{3}})\)

\(y = 10^{2\frac{1}{3}} ⋅ x^{-\frac{1}{3}}\)
\(y = 215 ⋅ x^{-0{,}33}\)

De grafiek hoort bij een lineair verband.

Hierbij hoort de formule \(y = a x + b \text{.}\)