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<p>Prior to this unit, students have studied what it means for a relationship to be a function, used function notation, and investigated linear and exponential functions. In this unit, they begin by looking at patterns that grow quadratically. They contrast this growth with linear and exponential growth. They further observe that eventually these quadratic patterns grow more quickly than linear patterns but more slowly than exponential patterns.</p>
<p>Students examine real-life examples of free-falling objects whose height over time can be modeled with quadratic functions. They use tables, graphs, and equations to describe the movement of these objects, eventually looking at situations where a projectile is launched upward. This leads to the important interpretation that in a quadratic function such as \(f(t)=5+30t&minus;16t^2\), representing the vertical position of an object after \(t\) seconds, 5 represents the initial height of the object, \(30t\) represents its initial upward path, and \(-16t^2\) represents the effect of gravity. Through this investigation, students also begin to appreciate how the different coefficients in a quadratic function influence the shape of the graph. In addition to projectile motion, students examine other situations represented by quadratic functions, including area and revenue.</p>
<p>Next, students examine the standard and factored forms of quadratic expressions. They investigate how each form is useful for understanding the graph of the function defined by these equivalent forms. The factored form is helpful for finding when the quadratic function takes the value 0 to obtain the \(x\)-intercept(s) of its graph, while the constant term in the standard form shows the \(y\)-intercept. Students also find that the factored form is useful for finding the vertex of the graph because its \(x\)-coordinate is halfway between the points where the graph intersects the \(x\)-axis (if it has two \(x\)-intercepts). As for the standard form, students investigate the coefficients of the quadratic and linear terms further, noticing that the coefficient of the quadratic term determines if it opens upward or downward. The effect of the coefficient of the linear term is somewhat mysterious and more complicated. Students explore how it shifts the graph both vertically and horizontally in an extension lesson.</p>
<p>Finally, students investigate the vertex form of a quadratic function and understand how the parameters in the vertex form influence the graph. They learn how to determine the vertex of the graph from the vertex form of the function. They also begin to relate the different parameters in the vertex form to the general ideas of horizontal and vertical translation and vertical stretch, ideas that will be investigated further in a later course.</p>
<p>Note on materials: Access to graphing technology is necessary for many activities. Examples of graphing technology are: a handheld graphing calculator, a computer with a graphing calculator application installed, and an internet-enabled device with access to a site like <a href="https://desmos.com/calculator" target="_blank">desmos.com/calculator</a> or <a href="https://geogebra.org/graphing" target="_blank">geogebra.org/graphing</a>. For students using the digital version of these materials, a separate graphing calculator tool isn&rsquo;t necessary. Interactive applets and Desmos graphing calculator frames are embedded throughout.</p>