Example 1
Completion requirements
| Example 1 |
For the quadratic function \(f(x) = 2(x + 2.5)^2 - 0.5 \),
Along with the graphs of quadratic functions being parabolas, there are other important characteristics to discuss. One key characteristic is the vertex, which is the highest or lowest point on the graph. A parabola can open upward, giving the vertex the lowest function value, or it can open downward, giving the vertex the highest function value. In Example 1, the vertex occurs at the lowest point on the curve, where the function reaches its lowest value. Also, there is a line of symmetry that passes through the vertex, which is called the axis of symmetry. Another characteristic of quadratic functions that differs from linear functions is that quadratic functions can have zero, one, or two \(x\)-intercepts. (You will see examples of functions with zero \(x\)-intercepts shortly.) The rest of Lesson 2.1 will examine the characteristics of quadratic functions .
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Complete a table of values for \(x\)-values from \(-5\) to \(0\).
\(x \) (input)
\(f(x)\) (output)
(\(x\), \(f(x)\)) \(–5 \)
\(f(-5) = 2(-5 + 2.5)^2 - 0.5 = 12\) \((–5, 12) \)
\(–4\) \(f(-4) = 2(-4 + 2.5)^2 - 0.5 = 4\) \((–4, 4) \)
\(–3\) \(f(-3) = 2(-3 + 2.5)^2 - 0.5 = 0\) \((–3, 0) \)
\(–2\) \(f(-2) = 2(-2 + 2.5)^2 - 0.5 = 0\) \((–2, 0)\) \(–1\) \(f(-1) = 2(-1 + 2.5)^2 - 0.5 = 4\) \((–1, 4) \)
\(0\) \(f(0) = 2(0 + 2.5)^2 - 0.5 = 12\) \((0, 12) \)
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Graph the function, drawing a smooth curve through the plotted points.
Window Settings XMin \(−6 \)
XMax \(1\) XScale
\(1\)
YMin \(−2\)
YMax \(13\)
YScale \(1\)
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Determine the \(x\)- and \(y\)-intercepts.
The function has \(x\)-intercepts at \(–3\) and \(–2\). The \(y\)-intercept is \(12\). -
Determine the domain and range of the function.
The domain is {\(x | x \in \rm{R}\)}.
To determine the range, first determine the lowest point on the graph, \((–2.5, –0.5)\). Therefore, the range is {\(y | y \ge –0.5,\thinspace y \in \rm{R}\)}. -
Draw a vertical line through the line of symmetry.
Note the red dashed vertical line passing through the vertex. This line is the axis of symmetry; the function is a mirror image of itself across this line, which has an equation of \(x = –2.5 \).
Along with the graphs of quadratic functions being parabolas, there are other important characteristics to discuss. One key characteristic is the vertex, which is the highest or lowest point on the graph. A parabola can open upward, giving the vertex the lowest function value, or it can open downward, giving the vertex the highest function value. In Example 1, the vertex occurs at the lowest point on the curve, where the function reaches its lowest value. Also, there is a line of symmetry that passes through the vertex, which is called the axis of symmetry. Another characteristic of quadratic functions that differs from linear functions is that quadratic functions can have zero, one, or two \(x\)-intercepts. (You will see examples of functions with zero \(x\)-intercepts shortly.) The rest of Lesson 2.1 will examine the characteristics of quadratic functions .