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Enter values to write the function that matches the graph shown.

Enter values to write the function that matches the graph shown.-example-1

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Answer:


p(x)=\left(\:\boxed{-3}\:x+6\right)\left(\:\boxed{4}\:x-4\right)

Explanation:

The x-intercepts are the points at which the curve crosses the x-axis.

From inspection of the given graph, the x-intercepts are:

  • x = 1
  • x = 2

As the parabola opens downwards, the leading coefficient is negative.


\boxed{\begin{minipage}{6 cm}\underline{Intercept form of a quadratic equation}\\\\$y=a(x-r_1)(x-r_2)$\\\\where:\\ \phantom{ww}$\bullet$ $r_1$ and $r_2$ are the $x$-intercepts. \\ \phantom{ww}$\bullet$ $a$ is some constant.\\\end{minipage}}

Therefore:


p(x)=-a(x-1)(x-2)

Expand:


\implies p(x)=-a(x^2-3x+2)


\implies p(x)=-ax^2+3ax-2a

The y-intercept is the point at which the curve crosses the y-axis, so when x = 0. As the curve intersects the y-axis at y = -24:


\begin{aligned}\implies p(0)=-a(0)^2+3a(0)-2a&=-24\\-2a&=-24\\a&=12\end{aligned}

Therefore the equation in intercept form is:


p(x)=-12(x-1)(x-2)

Rewrite -12 as 4 · -3:


\implies p(x)=4 \cdot -3(x-1)(x-2)

Multiply (x - 1) by 4 and (x - 2) by -3:


\implies p(x)=4(x-1)(-3)(x-2)


\implies p(x)=(4x-4)(-3x+6)

Applying the commutative property of multiplication:


\implies p(x)=(-3x+6)(4x-4)

Compare with the given function:


\implies p(x)=\left(\:\boxed{-3}\:x+6\right)\left(\:\boxed{4}\:x-4\right)

User VernonFuller
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