1Cademy - Example 9.61: Graphing $$f(x) = -2x^2

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Graphing a Quadratic Equation in Two Variables

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Example 9.61: Graphing $f(x) = -2x^2 - 4x + 2$

To graph the quadratic function $f(x) = -2x^2 - 4x + 2$ using transformations, first rewrite the function in vertex form, $f(x) = a(x - h)^2 + k$ , by completing the square. Start by factoring out the leading coefficient $-2$ from the $x$ terms: $f(x) = -2(x^2 + 2x) + 2$ . Next, complete the square inside the parentheses. Take half of the $x$ -coefficient ( $\frac{2}{2} = 1$ ) and square it to get $1$ . Add and subtract $1$ inside the parentheses: $f(x) = -2(x^2 + 2x + 1 - 1) + 2$ . Distribute the $-2$ to the $-1$ to move it outside the parentheses: $f(x) = -2(x^2 + 2x + 1) + 2 + 2$ , which simplifies to the vertex form $f(x) = -2(x + 1)^2 + 4$ . From this form, the transformations can be identified by comparing it to the basic parabola $f(x) = x^2$ . The $(x + 1)$ indicates a horizontal shift to the left by $1$ unit. The $-2$ multiplier indicates a vertical stretch by a factor of $2$ and a reflection across the $x$ -axis, causing the parabola to open downward. Finally, the $+ 4$ represents a vertical shift upward by $4$ units. Applying these transformations in sequence produces the graph of $f(x) = -2x^2 - 4x + 2$ with a vertex at $(-1, 4)$ .

Updated 2026-05-25

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OpenStax Intermediate Algebra

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