Question
Linear functions are ALWAYS increasing or decreasing but never both.
Quadratic functions will ALWAYS have one interval of increasing and one interval of decreasing. Response area
Exponential functions are always increasing. Response area
true or false
Quadratic functions will ALWAYS have one interval of increasing and one interval of decreasing. Response area
Exponential functions are always increasing. Response area
true or false
Answers
GPT-4o mini
Let's analyze the statements about each type of function:
1. **Linear functions:** This statement is **true**. A linear function has a constant slope, meaning it will either always increase (if the slope is positive) or always decrease (if the slope is negative). It cannot be both.
2. **Quadratic functions:** This statement is **true**. A standard quadratic function (like \(f(x) = ax^2 + bx + c\), where \(a \neq 0\)) is a parabola. If \(a > 0\), it opens upwards and has one interval of increasing (to the right of the vertex) and one interval of decreasing (to the left of the vertex). If \(a < 0\), it opens downwards and has one interval of decreasing (to the right of the vertex) and one interval of increasing (to the left of the vertex).
3. **Exponential functions:** This statement is also **true**. Exponential functions of the form \(f(x) = a \cdot b^x\) (where \(a > 0\) and \(b > 1\)) are always increasing for all \(x\) since the growth rate continuously increases. If \(0 < b < 1\), the function is always decreasing.
In summary, all three statements are true.
1. **Linear functions:** This statement is **true**. A linear function has a constant slope, meaning it will either always increase (if the slope is positive) or always decrease (if the slope is negative). It cannot be both.
2. **Quadratic functions:** This statement is **true**. A standard quadratic function (like \(f(x) = ax^2 + bx + c\), where \(a \neq 0\)) is a parabola. If \(a > 0\), it opens upwards and has one interval of increasing (to the right of the vertex) and one interval of decreasing (to the left of the vertex). If \(a < 0\), it opens downwards and has one interval of decreasing (to the right of the vertex) and one interval of increasing (to the left of the vertex).
3. **Exponential functions:** This statement is also **true**. Exponential functions of the form \(f(x) = a \cdot b^x\) (where \(a > 0\) and \(b > 1\)) are always increasing for all \(x\) since the growth rate continuously increases. If \(0 < b < 1\), the function is always decreasing.
In summary, all three statements are true.