Problem

Use the graphs and equations to compare \(L(x)~=~3x~+~1\) and \(Q(x)~=~x^{2}\) on all real inputs. State each function's type, rate behavior, extrema, interval behavior, and their key contrast.

Actual source graphs of the stated linear and quadratic functions without feature labels.

Big Picture

What this problem is really about

A line and a parabola differ most clearly in how their rates behave across the entire domain. We’ll connect the equations to the graphs, checking for constant versus changing rate, any turning point, and the intervals where each function rises or falls. Those features also tell us whether a global maximum or minimum can exist.

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Preview only

L = quadratic with changing rate, minimum (0,1), decreasing then increasing; Q = linear with constant slope 1, increasing for all x, no extremum; contrast = L turns while Q has constant rate

L = linear with constant slope 3, increasing for all x, minimum (0,1); Q = quadratic with changing rate, no extremum, increasing for all x; contrast = both have no turning point

L = linear with constant slope 3, increasing for all x, no global extremum; Q = quadratic with constant slope 3, increasing for all x, no extremum; contrast = both increase at the same constant rate

L = linear with constant slope 3, increasing for all x, no global extremum; Q = quadratic with changing rate, minimum (0,0), decreasing on (-∞,0] then increasing on [0,∞); contrast = L has constant rate while Q turns and its rate changes

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Use the graphs and equations to compare \(L(x)~=~3x~+~1\) and \(Q(x)~=~x^{2}\) on all real inputs. State each function's type, rate behavior, extrema, interval behavior, and their key contrast.

What this problem is really about

Turn the preview into practice.

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