Question

Write the equations of the linear and exponential functions that pass through the points (0,6) and (1,9)

Answer 1

Answer:

$f(x) = 3\, x + 6$.

$\displaystyle f(x) = 6\, \left(\frac{3}{2}\right)^{x}$.

Step-by-step explanation:

A linear function in a cartesian plane can be written in the form:

$f(x) = m\, x + k$,

Where $m$ (slope) and $k$ ($y$-intercept) are constants.

Given the two points $(0,\, 6)$ and $(1,\, 9)$, obtain two equations of $m$ and $k$ by substituting in the value of $(x,\, f(x))$ at each point:

• $(0,\, 6)$: $f(x) = 6$ when $x = 0$. Hence, $6 = (m)\, (0) + k$, which simplifies to $k = 6$.
• $(1,\, 9)$: $f(x) = 9$ when $x = 1$. $9 = (m)\, (1) + k$. Substitute in $k = 6$ to obtain $m = 3$.

Hence, the equation of this linear function would be:

$f(x) = 3\, x + 6$.

An exponential function in a cartesian plane can be written in the form:

$f(x) = a\, b^{x}$,

Where $a \ne 0$ and $b > 0$ are constants.

Similar to the example of the linear equation, the two points $(0,\, 6)$ and $(1,\, 9)$ provide two equations for $a$ and $b$:

• $(0,\, 6)$: $6 = a\, b^{0}$. Since $b > 0$, $b^{0} = 1$. Hence, this equation simplifies to $a = 6$.
• $(1,\, 9)$: $9 = a\, b^{1}$. Substitute in $k = 6$ to obtain $b = (3/2)$.

Hence, the equation of this exponential function would be:

$\displaystyle f(x) = 6\, \left(\frac{3}{2}\right)^{x}$.