Equivalent Forms Worksheets Grade 9

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is circumscribed around the 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. (2x² - 5x + 3)(x - 2) = ?
3. Charlotte is analyzing the trajectory of a drone during a search-and-rescue operation. The drone's height in meters above the ground at time t seconds is given by the quadratic function h(t) = -4t² + 32t + 2. Charlotte needs to rewrite this function in vertex form to determine the maximum height the drone reaches and when it occurs. Which equivalent form of the function reveals these properties, and what are the coordinates of the vertex?

…and 4 more problems

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,2x). The hypotenuse of this triangle has length 5√5 units. What is the value of x?
2. Isabella is designing a parabolic arch for a new entrance to a botanical garden. The shape of the arch can be modeled by the quadratic function h(x) = -x² + 7x - 12, where h(x) is the height in meters at a horizontal distance x meters from the left base. The construction team needs to know the horizontal distances where the arch meets the ground (the zeros of the function). Rewrite h(x) in its factored form, and explain which form — the original standard form or the factored form — more clearly reveals the zeros of the function. Then, state the two x-values where the arch touches the ground.
3. 2x² - 8x + 6 = ? when x = 5

…and 4 more problems

1. Ava draws a parabola that passes through the points (7, 0) and (13, 0) and has its vertex at (10, -9). Which of the following equivalent forms of the quadratic function best reveals the y-intercept of the parabola, and what is the y-intercept? Form A: y = (x - 10)^2 - 9 Form B: y = (x - 7)(x - 13) Form C: y = x^2 - 20x + 91
2. A drone is flying along a path modeled by the quadratic function h(t) = -2t² + 12t + 5, where h represents the drone's height in meters above ground level and t represents time in seconds. The drone operator wants to know the maximum height the drone reaches during its flight. What is the maximum height in meters?
3. Mere graphs two equivalent quadratic functions on a coordinate plane. The first function is written in vertex form: f(x) = (x - 12)^2 + 15. The second function is written in standard form: g(x) = x^2 - 24x + 159. Which form reveals the coordinates of the vertex directly, and what is the vertex? Explain which form is more useful for identifying the axis of symmetry.

…and 4 more problems