Equivalent Forms Worksheets Grade 9

1. A research scientist is studying the decay of a radioactive isotope. The amount remaining after t years is modeled by the exponential function A(t) = 800 × (1/2)^(t/12). The scientist needs to determine when only 100 grams of the isotope will remain. How many years will it take for the isotope to decay to this amount?
2. A construction company is designing a suspension bridge where the cable shape follows a parabolic curve. The cable's height above the roadway is modeled by the function h(x) = -0.02x² + 0.8x + 10, where x is the horizontal distance in meters from the left tower and h(x) is the height in meters. The engineers need to find the horizontal distance from the left tower where the cable reaches its maximum height. At what distance x does the cable reach its maximum height?
3. A tech company is modeling the depreciation of their computer equipment using the function V(t) = 1200 × (0.85)^t, where V(t) represents the value in dollars after t years. Their accountant needs to rewrite this function in the form V(t) = 1200 × e^(kt) to match their financial software. Which expression is equivalent to the original depreciation function?

…and 4 more problems

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (0,6). A circle is circumscribed around this triangle such that all three vertices lie on the circle's circumference. What is the area of the circumscribed circle? (Use π = 3.14)
2. (3x² - 5x + 2) + (2x² + 4x - 7) = ?
3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A circle is inscribed within this triangle, tangent to all three sides. What is the radius of this inscribed circle?

…and 6 more problems

1. Liam is designing a rectangular garden with a perimeter of 60 meters. He wants to model the area of the garden as a function of its width. If the width is represented by w meters, what is the area function A(w) in standard quadratic form?
2. (3x² - 5x + 2) + (2x² + 7x - 4) = ?
3. Which form of the quadratic function y = x² - 14x + 49 reveals that it has a double root at x = 7?

…and 6 more problems