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

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=-x^{2}\\&g(x)=-x^{2}+1\\&h(x)=-(x+3)^{2}\end{aligned}$$.

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, represented in forms such as ax² + bx + c or the vertex form a(x - h)² + k. Their graphs are parabolas, which have a vertex and an axis of symmetry, and the sign of the coefficient a determines whether they open upward or downward.

Graph Transformations

Graph transformations involve modifying the base graph of a function by applying operations such as translations, reflections, stretches, or compressions. These techniques allow you to easily sketch related functions by understanding how shifts or reflections affect the graph.

Vertical Shifts

A vertical shift occurs when a constant is added to or subtracted from the function, moving the graph up or down along the y-axis without changing its overall shape. This is used to adjust the position of the vertex for functions altered by a constant term.

Horizontal Shifts

A horizontal shift happens when a constant is added to or subtracted from the input variable, moving the graph left or right along the x-axis. This translation changes the x-coordinate of the vertex or the center of symmetry in quadratic functions.

Reflections

Reflections flip the graph of a function across a line, typically the x-axis or y-axis. In quadratic functions, a negative coefficient in front of the squared term produces a vertical reflection, changing the direction in which the parabola opens.

Transcript

00:01 In this problem we are given 3 functions.

00:04 First, we have f of x equals minus x square.

00:08 The next function is g of x that is equal to minus x squared plus 1.

00:14 So we can write this as f of x plus 1.

00:18 And the third function is h of x which is equal to minus x plus 3 whole square.

00:25 So we can write this as we know minus x square is the function f of x.

00:30 So this is the function f of x plus 3.

00:36 Now to plot these three functions, we will first sketch them by hand and then verify it using a graphing calculator.

00:43 So first we draw f of x.

00:46 Now we know f of x equals x square would be the parent function of any parabola.

00:51 Now because we have a minus here, this will just be the mirror image along the x axis.

00:58 So it will be a downwards facing parabola like this.

01:00 This will be f of x.

01:03 Its vertex will be at the origin.

01:06 Next we have g of x which is written as f of x plus 1.

01:10 Now because the function value is added with 1, that means this will be the same graph but stretched upwards or shifted upwards rather by one step.

01:21 So its vertex should lie at the point 0 .1...

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