The death rates f (per 100,000 people) attributed to heart...

The death rates $f$ (per 100,000 people) attributed to heart disease in the United States from 1985 to 2003 can be modeled by the equation $f(t)=0.064 t^{2}-9.30 t+416.5, \quad 5 \leq t \leq 23$ 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 death rates $f$ (per 100,000 people) attributed to heart disease in the United States from 1985 to 2003 can be modeled by the equation
$f(t)=0.064 t^{2}-9.30 t+416.5, \quad 5 \leq t \leq 23$
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

Quadratic Functions in Standard Form

A quadratic function is typically written as ax² + bx + c, which clearly shows the coefficients affecting the shape and position of its parabola. Writing a quadratic in standard form aids in identifying features like the vertex and axis of symmetry, and it simplifies the process of analyzing and graphing the function.

Graphing Functions Using Technology

Graphing utilities or software are essential tools for visualizing mathematical functions. They enable users to see how transformations, such as shifts or stretches, impact the graph and to compare different models simultaneously. This practice helps in verifying algebraic manipulations and deepening one’s understanding of function behavior.

Horizontal Function Shifts

This concept involves moving a graph left or right by modifying the independent variable in a function. Replacing t with t + c shifts the graph to the left by c units, while t - c shifts it to the right by c units. Understanding this transformation is key when adjusting models to different reference points or time frames.

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