Analyzing Quadratic Graphs, in Depth
Read this topicProblems 10
- (ARML past Individual Round problem) A quadratic function f(x)f(x)f(x) with 222 as a root is such that 555 is the only real root of f(f(x)).f(f(x)).f(f(x)). Compute f(0).f(0).f(0).
- What is the maximum value of 3x2+9x+173x2+9x+7\frac{3x^2+9x+17}{3x^2+9x+7}3x2+9x+73x2+9x+17 over all real numbers x?x?x? Source: ARML 1986
- What is the minimum value of the quadratic x2+6x+17x^2 + 6x + 17x2+6x+17?
- A quadratic function y=ax2+bx+cy = ax^2+bx+cy=ax2+bx+c passes through the points (1,h)(1,h)(1,h) and (−1,k)(-1,k)(−1,k). If h−k=12h-k = 12h−k=12, what is bbb?
- At what value of xxx does the quadratic function f(x)=(x−3)2+(x−5)2f(x) = (x-3)^2 + (x-5)^2f(x)=(x−3)2+(x−5)2 attain its minimum value?
- Let f(x)=x2+x+1f(x) = x^2+x+1f(x)=x2+x+1. What is the minimum possible value of f(1−u)+f(1+u)f(1-u) + f(1+u)f(1−u)+f(1+u)?
- What are the xxx intercepts for f(x)=3x2+5x−8f(x)=3x^2+5x-8f(x)=3x2+5x−8?
- What are the xxx intercepts for f(x)=2x2−10x+13f(x)=2x^2-10x+13f(x)=2x2−10x+13?
- What is the axis of symmetry for f(x)=−6x2+2x+5f(x)=-6x^2+2x+5f(x)=−6x2+2x+5?
- What is the axis of symmetry for f(x)=−4x2−x+9f(x)=-4x^2-x+9f(x)=−4x2−x+9?