Quadratic functions: แบบฝึกหัด

Express the quadratic function \( f(x) = 2x^2 - 12x + 7 \) in the form \( a(x - h)^2 + k \). State the coordinates of the minimum point.

Find the values of the constant \( p \) such that the line \( y = px - 1 \) is a tangent to the curve \( y = x^2 - 4x + 3 \).

A quadratic function is given by \( y = -x^2 + 4x + 5 \). By completing the square, determine the maximum value of the function.

The path of a projectile is modeled by the function \( h(t) = -5t^2 + 20t + 2 \), where \( h \) is the height in meters and \( t \) is the time in seconds. Find the maximum height reached by the projectile.

Find the range of values of \( k \) for which the quadratic function \( f(x) = x^2 + 6x + k \) is always positive for all real values of \( x \).

Find the maximum value of the quadratic function \( f(x) = 11 + 6x - x^2 \) by using the method of completing the square.

By completing the square, find the minimum value of the quadratic function \(f(x) = x^2 - 6x + 13\) and the value of \(x\) at which it occurs.

Find the range of values of the constant \( k \) for which the quadratic function \( f(x) = (k-1)x^2 + 2kx + 3 \) is always positive for all real values of \( x \).

A quadratic function is defined by \(f(x) = -2x^2 + 12x - 10\).
(a) Express \(f(x)\) in the form \(a(x-h)^2 + k\).
(b) State the coordinates of the maximum point of the graph of \(y = f(x)\).
(c) Find the coordinates of the points where the graph intersects the \(x\)-axis.

A ball is thrown from the top of a building. Its height, \( H \) metres, above the ground \( t \) seconds after being thrown is modeled by the quadratic function \( H(t) = -5t^2 + 20t + 25 \).
(a) Find the maximum height reached by the ball.
(b) Calculate the time taken for the ball to reach the ground.
(c) Determine the range of values of \( t \) for which the height of the ball is at least 40 metres.