Find two quadratic functions, one that opens upward and one that opens downward, whose graphs ha

Find two quadratic functions, one that opens upward and one...

Key Concepts

Leading Coefficient and Parabola Orientation

The leading coefficient 'a' in a quadratic function determines the direction in which the parabola opens. If a is positive, the parabola opens upward; if a is negative, the parabola opens downward. This property is crucial for modifying a quadratic function to match specific requirements regarding its orientation.

Factored Form of a Quadratic Function

The factored form of a quadratic function is expressed as f(x) = a(x - r)(x - s), where r and s are the x-intercepts of the function. This form directly shows the values at which the function will be zero and is particularly useful when the roots are known. It also makes it easy to adjust the function by applying different values for the leading coefficient a.

X-Intercepts (Roots) of a Quadratic Function

The x-intercepts of a quadratic function, also known as the roots or zeros, are the values of x at which the function equals zero (f(x) = 0). These intercepts represent the points where the graph of the function crosses the x-axis. Finding these intercepts is often done by factoring, completing the square, or using the quadratic formula.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a is non-zero. The graph of a quadratic function is a parabola, which may open upward or downward depending on the sign of the leading coefficient.

Transcript

00:01 Hi everybody, so today we're going to be finding and building our own quadratic functions, and we're going to do it twice.

00:09 We want to find two quadratic functions that pass through these x intercepts of 0, 0, and 10, 0, and we want one that opens up and one that opens down.

00:24 So to start off, we're just going to start off with our most basic quadratic equation.

00:29 Y is equal to x, squared.

00:33 As you can see, that already passes through 1 .0, 0, 0, but we need to pass through a second.

00:39 So we're going to start by shifting our vertex to the very center.

00:44 And to do that, we're going to do x minus 5.

00:51 And this minus is very important because remember it moves the opposite of what you think it would.

00:56 Subtracting it, moves it to the right, and adding it, moves it to the left.

01:00 So now we have a vertex in the very center of our two points.

01:07 Now we just need to move it down enough.

01:11 So to start, we're going to go and write our new equation out in linear form.

01:21 So that'll be x squared minus 10x plus 25.

01:29 As you can see, it gives us the same equation, we're still high.

01:34 And we just want to move it down.

01:39 And to do that, i'm just going to go ahead and subtract 25.

01:44 That shifts the equation down, and gives us these two points, 0 .0 and 10 .0...

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