9th Grade

{
"voice_prompt": "",
"manuscript": {
"title": {
"text": "Why Do Exponential Functions Grow by Equal Factors Over Equal Intervals?"
},
"description": {
"text": "Because that’s what exponential functions do, they multiply by the same number every time x takes a step.",
"audio": "Because that’s what exponential functions do, they multiply by the same number every time x takes a step."
},
"scenes": [
{
"text": "Let's start with an example: f of x equals 3 times 2 to the power of x. Watch how the values change as x increases.",
"latex": "f(x) = 3 \\times 2^x"
},
{
"text": "When x is 0, f of x is 3. At x equals 1, it becomes 6. Then 12, 24, 48, and so on. Each time x increases by 1, f of x doubles. That’s constant growth by a factor.",
"latex": "f(0) = 3,\\ f(1) = 6,\\ f(2) = 12,\\ f(3) = 24,\\ f(4) = 48"
},
{
"text": "This is what defines exponential functions. They don’t grow by adding — they grow by multiplying. That’s what sets them apart from linear functions, which increase by the same amount each time. You can also see this in the graph. The curve gets steeper with each step. That’s what exponential growth looks like.",
"latex": "a+b"
},
{
"text": "Now let’s show this more generally. Suppose f of x equals a times b to the x, where a is the starting value and b is the growth factor.",
"latex": "f(x) = a \\times b^x"
},
{
"text": "Now compare two values: f of x, and f of x plus k, where k is the size of your interval. f of x equals a times b to the x, and f of x plus k equals a times b to the power of x plus k.",
"latex": "f(x) = a \\times b^x \\quad \\text{and} \\quad f(x + k) = a \\times b^{x + k}"
},
{
"text": "Next, divide f of x plus k by f of x. This shows how much the function grows over the interval of length k. f of x plus k divided by f of x equals a times b to the x plus k, over a times b to the x.",
"latex": "\\frac{f(x + k)}{f(x)} = \\frac{a \\times b^{x + k}}{a \\times b^x}"
},
{
"text": "The a's and b to the x cancel out. You are left with: f of x plus k divided by f of x equals b to the k.",
"latex": "\\frac{f(x + k)}{f(x)} = b^k"
},
{
"text": "So for every increase of x by k units, the function grows by a factor of b to the k.",
"latex": "b^k"
},
{
"text": "This is true for any exponential function. Equal steps in x always lead to equal growth factors, because of how the formula works. That’s why exponential functions appear in science, money, and health. Things like viruses, investments, and populations all grow by multiplying, not by adding.",
"latex": "a+c"
}
],
"outro": {
"text": "Exponential functions grow by multiplying, not adding. That’s why they increase by the same factor over every equal step.",
"audio": "Exponential functions grow by multiplying, not adding. That’s why they increase by the same factor over every equal step."
}
}
}