Quadratic functions: แบบฝึกหัด

Find the solution set of the inequality \(x^2 - 7x + 10 \le 0\).

Find the range of the function \(f(x) = x^2 - 4x + 7\) for the domain \(0 \le x \le 5\).

The line \(y = x + c\) is a tangent to the curve \(y = x^2 - 3x + 5\). Find the value of \(c\).

Find the coordinates of the turning point of the curve \(y = x^2 + 6x + 8\).

Find the range of values of \(k\) for which the equation \(x^2 + (k+1)x + 9 = 0\) has no real roots.

Write down the discriminant of the quadratic equation \( 2x^2 - 5x + 3 = 0 \) and state the nature of its roots.

Find the coordinates of the points of intersection between the line \( y = 2x + 1 \) and the curve \( y = x^2 - 2 \).

A curve has the equation \( y = kx^2 - 4x + (k-3) \), where \( k \) is a constant. Find the set of values of \( k \) for which the curve lies entirely below the \( x \)-axis.

The quadratic function \( f(x) \) is defined by \( f(x) = 2x^2 - 12x + 19 \).
(a) Express \( f(x) \) in the form \( a(x-h)^2 + k \).
(b) State the coordinates of the minimum point of the graph of \( y = f(x) \).
(c) State the range of \( f(x) \) for all real values of \( x \).
(d) Sketch the graph of \( y = f(x) \), labelling the minimum point and the y-intercept.

Find the range of values of the constant \(k\) for which the line \(y = 2x + k\) intersects the curve \(y = x^2 - 4x + 1\) at two distinct points.