Activity 1.5.3.
Reason algebraically using appropriate properties of quadratic functions to answer the following questions. Use Desmos to check your results graphically.
(a)
How many quadratic functions have \(x\)-intercepts at \((-5,0)\) and \((10,0)\) and a \(y\)-intercept at \((0,-1)\text{?}\) Can you determine an exact formula for such a function? If yes, do so. If not, explain why.
(b)
Suppose that a quadratic function \(q\) has vertex \((-3,-4)\) and opens upward. How many \(x\)-intercepts can you guarantee the function has? Why?
(c)
In addition to the information in (b), suppose you know that \(q(-1) = -3\text{.}\) Can you determine an exact formula for \(q\text{?}\) If yes, do so. If not, explain why.
(d)
Does the quadratic function \(p(x) = -3(x+1)^2 + 9\) have \(0\text{,}\) \(1\text{,}\) or \(2\) \(x\)-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist.
(e)
Does the quadratic function \(w(x) = -2x^2 + 10x - 20\) have \(0\text{,}\) \(1\text{,}\) or \(2\) \(x\)-intercepts? Reason algebraically to determine the exact values of any such intercepts or explain why none exist.
