State whether the product xy is negative or positive, and why.
x,y both positive
x positive, y negative
x,y both negative
What is the minimum value of \( x^2 \)?
Minimum of \( (x - 1)^2 \)? Hint: \(x -1\) takes all the same values as \(x\).
Minimum of \( (x - 1)(x + 1) \)? Hint: expand.
Let a,b be real numbers. Find the minimum of \( ax^2 + bx \) in x. ("Find the minimum in x" means to treat the other variables as constant, and find the value of x for which the function is minimized.)
What does the minimum of a quadratic expression tell us about the existence of real zeros for that expression?
To "find the zeros" of a quadratic expression is to find the value/s of x for which the equation \( ax^2 + bx + c = 0 \) holds. Once you know how to do this, you can solve an equation of the form \( ax^2 + bx + c = d \), where d is a constant, with no new techniques. Why is this so? Draw a to illustrate your point.
Use the app to experiment and develop an intuition for coefficients vs. shape, but answer the questions with exact answers, not estimates from the graph.
Fix \( a=1 \) . For what values of \( b \) and \( c \) does the curve have two zeros?
Fix \( c=-1, b=1 \). For what value/s of \( a \) does the curve have one zero?
The curve has two zeros. What can be said about \( a, b, \) and \( c \) ?