Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Veri

Sketch the graphs of the three functions by hand on the same...

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two that graph as parabolas. Their general form, a(x - h)² + k, shows that the vertex is at (h, k) and they open upward if a > 0 or downward if a < 0. Understanding the parabolic shape, axis of symmetry, and key points such as the vertex and intercepts is essential for analyzing and sketching these graphs by hand.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For quadratic functions, vertical shifts (adding or subtracting a constant) move the graph up or down, and horizontal shifts (modifying the x variable inside the function) move it left or right. Reflections, such as multiplying by -1, invert the graph across the x-axis. These concepts allow one to modify the base graph of a quadratic function to obtain the equation’s graph.

Vertex Form of a Parabola

The vertex form of a quadratic function, expressed as a(x - h)² + k, makes it straightforward to identify the vertex of the parabola and apply transformations. This form highlights how changes in the parameter h shift the graph horizontally and how changes in k shift it vertically. It also shows that multiplying by a affects the 'width' of the parabola and its direction of opening, which is key when sketching and comparing multiple quadratic functions on the same coordinate system.

Transcript

00:02 Given the functions f of x equals negative x squared, g of x equals negative x squared plus one, and h of x equals negative x minus 2 squared, we're going to graph these functions and see how they differ from one another.

00:15 This negative sign out front, it's actually at the beginning of each and every one of these functions, means that this function is reflected across the x axis.

00:25 So the parent function here is actually, we'll call it j of x, so that we use a different letter, equals x squared.

00:36 And it's a u -shaped parabola that looks something like this.

00:40 But we've got that negative sign out front, so it's going to flip it upside down.

00:45 So zero squared is zero.

00:48 It's not positive or negative, so it's going to stay at zero, zero.

00:51 And then if i put in a one, one squared is one, but we've got that negative out in front, so it's going to be negative one.

00:58 Negative 1 squared is 1 with the negative out front we get 1.

01:03 2 squared is 4 but there's a negative out front so we're going to be at negative 4 and the same thing with negative 2.

01:11 3 squared is 9 but it's negative so it's going to be down here at negative 9 and same thing with negative 3.

01:19 And when we graph that we'll see that upside down parabola or that upside down to you.

01:31 G of x equals negative x squared plus 1 so it's still reflected across the the x -axis, meaning it's upside down, so to speak.

01:40 And then this plus one is going to move everything up one spot.

01:44 So from here, i'm going to go up one.

01:47 I'm just going to go up one from each point.

01:57 And then when we graph that, we get this u -shaped parabola again.

02:03 Just shifted up one unit.

02:05 And on our last example, h of x there, we have h of x equals x minus two square.

02:12 This is going to shift everything two units, to the right...

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