The average tuition, room, and board costs f (in dollars)...

The average tuition, room, and board costs $f$ (in dollars) for undergraduates at public institutions from 1985 through 2005 can be approximated by the model $f(t)=6.22 t^{2}+115.2 t+2730, \quad 5 \leq t \leq 25$ where $t$ represents the year, with $t=5$ corresponding to 1985. (a) Adjust the model so that $t=0$ corresponds to 2000 rather than $t=5$ corresponding to $1985 .$ To do this, shift the graph of $f 20$ units to the left and obtain $g(t)=f(t+20) .$ Write $g(t)$ in standard form. (b) Use a graphing utility to graph $f$ and $g$ in the same viewing window.

The average tuition, room, and board costs $f$ (in dollars) for undergraduates at public institutions from 1985 through 2005 can be approximated by the model
$f(t)=6.22 t^{2}+115.2 t+2730, \quad 5 \leq t \leq 25$
where $t$ represents the year, with $t=5$ corresponding to 1985.
(a) Adjust the model so that $t=0$ corresponds to 2000 rather than $t=5$ corresponding to $1985 .$ To do this, shift the graph of $f 20$ units to the left and obtain $g(t)=f(t+20) .$ Write $g(t)$ in standard form.
(b) Use a graphing utility to graph $f$ and $g$ in the same viewing window.

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Key Concepts

Function Transformations

A function transformation involves altering the graph of a function through operations such as translations, reflections, dilations, or rotations. In this context, a horizontal shift is applied by replacing the variable t with t + 20, which moves the original graph 20 units to the left. This technique is key when adjusting models to change the reference point for the independent variable.

Standard Form of a Polynomial Function

The standard form of a quadratic function is given by at² + bt + c. Converting a function to this form requires expanding any expressions and combining like terms. Writing the transformed function in standard form makes it easier to analyze key characteristics such as the vertex, axis of symmetry, and intercepts.

Graphing Functions with Technology

Graphing both the original and transformed functions using a graphing utility allows for visual verification of the transformation. By plotting the functions in the same viewing window, one can observe the effects of shifting or other modifications, which aids in understanding the relationship between the analytical representation and its graphical depiction.

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