Exemplo 2 de Algebra and Trigonometry, 5.1 Quadratic Functions

Writing the Equation of a Quadratic Function from the Graph

Problema e solução em inglês, como no livro original.

Write an equation for the quadratic function $g$ in Figure 7 as a transformation of $f (x) = x^{2},$ and then expand the formula, and simplify terms to write the equation in general form.

Graph of a parabola with its vertex at (-2, -3).

Figure 7

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

Solução
We can see the graph of g is the graph of $f (x) = x^{2}$ shifted to the left 2 and down 3, giving a formula in the form $g (x) = a {(x - (−2))}^{2} - 3 = a {(x + 2)}^{2} - 3.$
Substituting the coordinates of a point on the curve, such as $(0, −1),$ we can solve for the stretch factor.

$\begin{matrix} - 1 & = & a {(0 + 2)}^{2} - 3 \\ 2 & = & 4 a \\ a & = & \frac{1}{2} \end{matrix}$

In standard form, the algebraic model for this graph is $(g) x = \frac{1}{2} {(x + 2)}^{2} - 3.$

To write this in general polynomial form, we can expand the formula and simplify terms.

$\begin{matrix} g (x) & = & \frac{1}{2} {(x + 2)}^{2} - 3 \\ = & \frac{1}{2} (x + 2) (x + 2) - 3 \\ = & \frac{1}{2} (x^{2} + 4 x + 4) - 3 \\ = & \frac{1}{2} x^{2} + 2 x + 2 - 3 \\ = & \frac{1}{2} x^{2} + 2 x - 1 \end{matrix}$

Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.

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A coordinate grid has been superimposed over the quadratic path of a basketball in Figure 8. Find an equation for the path of the ball. Does the shooter make the basket?

Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.

Figure 8 (credit: modification of work by Dan Meyer)

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O Try It do livro logo depois deste exemplo: mesma ideia, outros números. Só a resposta é dada.

A coordinate grid has been superimposed over the quadratic path of a basketball in Figure 8. Find an equation for the path of the ball. Does the shooter make the basket?

Figure 8 (credit: modification of work by Dan Meyer)

Solução

Sua vez

Solução

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