Math / Algebra

QuestionFind the equation for the linear function that passes through the points $(3,7)$ and $(5,19)$ . $\square$ Round all answers to 4 decimal places as needed Find the equation for the exponential function that passes through the points $(3,7)$ and $(5,19)$ . $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line and the equation of an exponential function, both going through two given points. Watch out! Don't mix up linear and exponential functions!
Remember, linear functions have a constant rate of change, while exponential functions have a constant ratio of change!

STEP 2

1. Tackle the Linear Function
2. Conquer the Exponential Function

STEP 3

Alright, for a linear function, we're looking for an equation of the form $y = mx + b$ .
Remember, $m$ is our slope (how steep the line is) and $b$ is our y-intercept (where the line crosses the y-axis).

STEP 4

Calculate the slope: The slope, $m$ , is the change in $y$ divided by the change in $x$ .
With our points $(3,7)$ and $(5,19)$ , this becomes: $m = \frac{19 - 7}{5 - 3} = \frac{12}{2} = 6$ Our slope is 6!
That means for every 1 unit we move to the right, we move up 6 units!

STEP 5

Find the y-intercept: Now that we have our slope, we can plug it into our equation along with one of our points.
Let's use $(3,7)$ : $7 = 6 \cdot 3 + b$ $7 = 18 + b$ Subtract 18 from both sides: $b = 7 - 18 = -11$ So, our y-intercept is -11!

STEP 6

Write the equation: Putting it all together, our linear equation is $y = 6x - 11$ !

STEP 7

Now, for the exponential function, we're looking for something of the form $y = ab^x$ .
Here, $a$ is our initial value (what $y$ is when $x=0$ ) and $b$ is our base (what we multiply by as $x$ increases).

STEP 8

Set up a system of equations: We can use our points to create two equations: $7 = ab^3$ $19 = ab^5$

STEP 9

**Solve for a and b:** A slick way to solve this is to divide the second equation by the first: $\frac{19}{7} = \frac{ab^5}{ab^3}$ The $a$ values divide to one, and we're left with: $\frac{19}{7} = b^2$ Take the square root of both sides (we only need the positive root since the base of an exponential function must be positive): $b = \sqrt{\frac{19}{7}} \approx 1.6482$ Now, plug this value of $b$ back into one of our original equations.
Let's use the first one: $7 = a(1.6482)^3$ $7 \approx a \cdot 4.4644$ Divide both sides by 4.4644: $a \approx \frac{7}{4.4644} \approx 1.5678$

STEP 10

Write the equation: Our exponential equation is approximately $y = 1.5678 \cdot (1.6482)^x$ !

STEP 11

Linear function: $y = 6x - 11$ Exponential function: $y = 1.5678(1.6482)^x$

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QuestionFind the equation for the linear function that passes through the points $(3,7)$ and $(5,19)$ . $\square$ Round all answers to 4 decimal places as needed Find the equation for the exponential function that passes through the points $(3,7)$ and $(5,19)$ . $\square$

Studdy Solution

STEP 1

STEP 2

1. Tackle the Linear Function
2. Conquer the Exponential Function

STEP 3

Alright, for a linear function, we're looking for an equation of the form $y = mx + b$ .
Remember, $m$ is our slope (how steep the line is) and $b$ is our y-intercept (where the line crosses the y-axis).

STEP 4

Calculate the slope: The slope, $m$ , is the change in $y$ divided by the change in $x$ .
With our points $(3,7)$ and $(5,19)$ , this becomes: $m = \frac{19 - 7}{5 - 3} = \frac{12}{2} = 6$ Our slope is 6!
That means for every 1 unit we move to the right, we move up 6 units!

STEP 5

Find the y-intercept: Now that we have our slope, we can plug it into our equation along with one of our points.
Let's use $(3,7)$ : $7 = 6 \cdot 3 + b$ $7 = 18 + b$ Subtract 18 from both sides: $b = 7 - 18 = -11$ So, our y-intercept is -11!

STEP 6

Write the equation: Putting it all together, our linear equation is $y = 6x - 11$ !

STEP 7

Now, for the exponential function, we're looking for something of the form $y = ab^x$ .
Here, $a$ is our initial value (what $y$ is when $x=0$ ) and $b$ is our base (what we multiply by as $x$ increases).

STEP 8

Set up a system of equations: We can use our points to create two equations: $7 = ab^3$ $19 = ab^5$

STEP 9

STEP 10

Write the equation: Our exponential equation is approximately $y = 1.5678 \cdot (1.6482)^x$ !

STEP 11

Linear function: $y = 6x - 11$ Exponential function: $y = 1.5678(1.6482)^x$

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Studdy Solution

STEP 1

STEP 2

1. Tackle the Linear Function2. Conquer the Exponential Function

STEP 3

Alright, for a linear function, we're looking for an equation of the form y=mx+by = mx + by=mx+b.Remember, mmm is our **slope** (how steep the line is) and bbb is our **y-intercept** (where the line crosses the y-axis).

STEP 4

STEP 5

STEP 6

**Write the equation:** Putting it all together, our linear equation is y=6x−11y = 6x - 11y=6x−11!

STEP 7

Now, for the exponential function, we're looking for something of the form y=abxy = ab^xy=abx.Here, aaa is our **initial value** (what yyy is when x=0x=0x=0) and bbb is our **base** (what we multiply by as xxx increases).

STEP 8

**Set up a system of equations:** We can use our points to create two equations: 7=ab3 7 = ab^3 7=ab3 19=ab5 19 = ab^5 19=ab5

STEP 9

STEP 10

**Write the equation:** Our exponential equation is approximately y=1.5678⋅(1.6482)xy = 1.5678 \cdot (1.6482)^xy=1.5678⋅(1.6482)x!

STEP 11

Linear function: y=6x−11y = 6x - 11y=6x−11 Exponential function: y=1.5678(1.6482)xy = 1.5678(1.6482)^xy=1.5678(1.6482)x

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1. Tackle the Linear Function
2. Conquer the Exponential Function

Alright, for a linear function, we're looking for an equation of the form $y = mx + b$ .
Remember, $m$ is our slope (how steep the line is) and $b$ is our y-intercept (where the line crosses the y-axis).

Write the equation: Putting it all together, our linear equation is $y = 6x - 11$ !

Now, for the exponential function, we're looking for something of the form $y = ab^x$ .
Here, $a$ is our initial value (what $y$ is when $x=0$ ) and $b$ is our base (what we multiply by as $x$ increases).

Set up a system of equations: We can use our points to create two equations: $7 = ab^3$ $19 = ab^5$

Write the equation: Our exponential equation is approximately $y = 1.5678 \cdot (1.6482)^x$ !

Linear function: $y = 6x - 11$ Exponential function: $y = 1.5678(1.6482)^x$