Fill in each blank with the appropriate word or phrase....

Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed. Discuss/Explain why the graph of a quadratic function has no $x$ -intercepts if $a$ and $k$ [vertex $(h, k)]$ have like signs. Under what conditions will the function have a single real root?

Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
Discuss/Explain why the graph of a quadratic function has no $x$ -intercepts if $a$ and $k$ [vertex $(h, k)]$ have like signs. Under what conditions will the function have a single real root?

Key Concepts

Parabola Orientation and x-intercepts

The sign of the coefficient a determines whether the parabola opens upward (a > 0) or downward (a < 0). When a and the y-coordinate of the vertex k have the same sign, the vertex is positioned on the same side of the x-axis as the opening direction of the parabola. Consequently, the entire graph will lie either above or below the x-axis, resulting in the absence of x-intercepts.

Condition for a Single Real Root

A quadratic function has a single real root, also known as a double root, when its graph touches the x-axis at exactly one point. This occurs when the discriminant (b² - 4ac) is zero, meaning the vertex of the parabola lies exactly on the x-axis. In vertex form, this corresponds to the specific scenario where k equals zero.

Vertex Form of a Quadratic Function

Quadratic functions can be expressed in vertex form as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form clearly shows the location of the vertex and helps in understanding the transformation of the parent quadratic function, indicating how the graph shifts and scales based on the values of h, k, and a.

Transcript

00:01 Hello, today we're talking about quadratic formulas and quadratic equations, and specifically talking about x intercepts or roots.

00:11 So an x intercept is where the quadratic equation crosses the x axis, for instance, like that.

00:18 There's x intercept there, x intercept there.

00:21 So this equation has two roots.

00:25 Now, if the leading coefficient, so in our quadratic equation, y equals a x squared plus b x plus c so a b and c here are all constants and their leading coefficient is a and if a is the same sign as the y value in the vertex so remember vertex is going to be at this point h and k are also just arbitrary variables for the specific x and y value of the extreme value of the quadratic equation.

01:14 So again, if this are quadratic equation, here's the minimum or extreme value, and that point is h comma k.

01:23 Now, if a and k are the same sign, then there won't be any x intercepts.

01:31 There will be no zeros in that equation because let's look at it on two graphs.

01:38 So if a is positive, then it's going to open upwards.

01:46 And so if a is positive and k is negative, then it's like what i showed before.

01:54 It's positive.

01:55 A is positive, so it opens upwards, k is negative, so there's your vertex.

02:01 So you're going to have your two x intersets.

02:05 And that's positive a, negative, k.

02:12 Now let's look at positive a, positive k.

02:20 So again, it's going to open upwards, but this time, that was bad.

02:28 So there we go...

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