Several graphs of the quadratic function ƒ(x) = ax^2 + bx + c are...

Question

Several graphs of the quadratic function ƒ(x) = ax^2 + bx + c are shown below. For the given restrictions on a, b, and c, select the corresponding graph from choices A–F. (Hint: Use the discriminant.) a < 0; ^b2 - 4ac < 0

Accepted Answer

Hey, everyone in this problem, we're asked which of the following graphs of a quadratic function satisfied the condition given below. We're given A is less than zero and B squared minus four AC is also less than zero. We're given four answer choices. They are all graphs. Option A and B show a Parabola opening upwards. Option A has two X intercepts or roots and option B has none. Option C and D show Parabola's opening downwards. Option C has no X intercepts and option D has one. Let's go back to the conditions we were given and we're gonna start by writing the equation for a quadratic function in general. OK. In standard form, we have Y is equal to A X squared plus B X plus C. Now we're told that A is less than zero and A is our leading coefficient. And so that's the coefficient corresponding to our X word term. Now, if A is less than zero that tells us that our parabola opens downwards. OK. So the para opens downwards which tells us that we're looking at option C or D and we can eliminate A and B. Now, the second piece of information we're given is that B squared minus four AC is less than zero. Now, B squared minus four AC recall that this is the discriminate. And where does that discriminate come from? Well, given our equation in standard form Y equals A X squared plus B X plus C. If we wanted to find the X intercepts or the roots of this equation, we could use the quadratic formula. It tells us that X is equal to negative B plus or minus the square root of B squared minus four ac divided by two A. And that discriminate is what's under the square root. So underneath that square root in the num numerator, we have B squared minus four AC. Now, in this case, we're told our discriminate is less than zero. So that means we're gonna have the square root of a negative number. Now, the square root of a negative number gives us no real solutions. That means that there are gonna be no real roots. OK? No real solutions for X, no real roots. And so our Parabola is going to have no real roots, OK? Or no real X intercepts. So using the two conditions we were given, we found that this Parabola opens downwards and then it has no real roots. So we've already eliminated auction A and B since they open upwards, if we go down to AC and D, they both open downwards, which is what we want. And we also found that this function has no real roots. And so this corresponds with answer choice C we have the graph of the parabola opening downwards that does not cross the X or the X axis. That's it for this one. Thanks everyone for watching. I hope this video helped.

Several graphs of the quadratic function ƒ(x) = ax^2 + bx + c are shown below. For the given restrictions on a, b, and c, select the corresponding graph from choices A–F. (Hint: Use the discriminant.) a < 0; ^b2 - 4ac < 0

Key Concepts

Quadratic Functions

Discriminant

Graphical Interpretation of Coefficients

Watch next