Quadratic Functions

Question 5: Diagram above shows the graphs of the curves y = x2 + x – kx + 5 and y = 2(x – 3) – 4h that intersect the x-axis at two points. Find (a) the value of k and of h, (b) the minimum value of each curve. Solution: (a) y= x 2 … Read more

Question 3: Given that the quadratic function f(x) = 2×2 – px + p has a minimum value of –18 at x = 1. (a) Find the values of p and q. (b) With the value of p and q found in (a), find the values of x, where graph f(x) cuts the x-axis. (c) … Read more

2.9.1 Quadratic Functions, SPM Practice (Long Questions) Question 1: Without drawing graph or using method of differentiation, find the maximum or minimum value of the function y = 2 + 4x – 3×2. Hence, find the equation of the axis of symmetry of the graph. Solution: By completing the square for the function in the … Read more

Question 5: Find the range of values of k if the quadratic equation 3(x2 – kx – 1) = k – k2 has two real and distinc roots. Solution: 3( x 2 −kx−1 )=k− k 2 3 x 2 −3kx−3−k+ k 2 =0 3 x 2 −3kx+ k 2 −k−3=0 a=3,b=−3k,c= k 2 −k−3 In cases of two real and distinc roots, … Read more

Question 3: The straight line y = 5x – 1 does not intersect with the curve y = 2×2 + x + h. Find the range of values of h. Solution: y=5x−1  …… (1) y=2 x 2 +x+h …… (2) Substitute (1) into (2), 5x−1=2 x 2 +x+h 2 x 2 +x+h−5x+1=0 2 x 2 −4x+h+1=0   b 2 −4ac<0 ( … Read more

2.8.1 Quadratic Functions, SPM Practice (Short Questions) Question 1: Find the minimum value of the function f (x) = 2×2 + 6x + 5. State the value of xthat makes f (x) a minimum value. Solution: By completing the square for f (x) in the form of f (x) = a(x + p)2 + q … Read more

Example 3Find the range of values of m for which the straight line y = m x + 6   does not meet the curve 2 x 2 − x y = 3  .

Example 2The straight line y = 2 k + 1   intersects the curve y = x + k 2 x   at two distinct points.  Find the range of values of k.

Example 1Find the value of p for which 8 y = x + 2 p   is a tangent to the curve 2 y 2 = x + p  .

Nature of the Roots (Combination of Straight Line and the Curve)When you have a straight line and a curve, you can solve the equation of the straight line and the curve simultaneously and form a quadratic equation, ax2 +bx + c = 0. The discriminant, b 2 − 4 a c  gives information about the number of … Read more