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.$$\begin{aligned}&f(x)=x^{2}\\&g(x)=3 x^{2}\\&h(x)=(x+2)^{2}+1\end{aligned}$$.

Key Concepts

Overlaying Multiple Graphs

Overlaying multiple graphs on a single coordinate system involves plotting different functions on the same set of axes. This method is useful for comparing the effects of various transformations, such as vertical scaling or translation, and allows for direct visual comparison of their properties.

Horizontal and Vertical Translation

Translation involves shifting the graph horizontally and/or vertically without changing its shape. A function of the form h(x) = (x + a)² + k moves the graph left by a units and up by k units. This technique helps in repositioning the vertex of the parabola, an essential aspect when analyzing the graph's features.

Quadratic Functions

Quadratic functions, represented by equations like f(x) = x², produce parabolic graphs that are symmetric about a vertical axis. They are fundamental in algebra and serve as the starting point for understanding more complex polynomial functions and their behaviors.

Vertical Scaling (Stretching and Compression)

Vertical scaling refers to multiplying the function by a constant, which stretches or compresses the graph along the y-axis. For instance, multiplying x² by 3 results in a narrower parabola, as the output values are increased, demonstrating how coefficients affect the curvature of quadratic graphs.

Transcript

00:01 In this question, the three given functions are f of x equal to x square.

00:08 Next we have g of x equal to 3x square which we can write as 3 times f of x and we have another function h of x which is equal to x plus 2 the whole square plus 1.

00:25 Now this we can write as we know f x is x square so here we can write this is f x square so here we can write this is f of x plus 2 and then because of this plus 1 we add a plus 1 so h of x can be written as f of x plus 1 so now let us catch these three graphs by hand first first we have to parent parabolic function f of x equal to x square which is simply a parabola passing through the origin it has its vertex at the origin so this is f of x the next function g of x is 3 times f of x which means it has been vertically stretched so the parabola will become thinner but its vertex will remain same so it will be a thinner parabola like this this will be g of x it is a vertically stretched version of the graph of f of x finally we have h of x now here first we have f of x plus 2 which means that the graph has been shifted towards the left by two steps and then there is a plus 1 with the function value which means the graph has been shifted upwards by one step.

01:39 So the vertex should come two steps to the left and one step higher, which means the vertex should be at this point minus 2 .1.

01:50 And it will be a parabola like this.

01:56 This should be h of x.

02:00 Now to verify this we will plot the same three functions using a graphing calculator.

02:05 So we'll use the ti -84 graphing calculator.

02:09 In here we'll first go to the y equals menu...

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