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6)  Determine para la función cuadrática $f(x)=x^2+6x+8$, los coeficientes $\ud835\udc4e$,$\ud835\udc4f$ \ud835\udc66 $\ud835\udc50$, su concavidad, corte en el eje Y, su eje de simetría, vértice y cortes en el eje X. Por medio de los elementos de la parábola construya la gráfica de $\ud835\udc53$.

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Para la función $f(x)=x^2+6x+8$ se tiene que sus coeficientes $a$, $b$ y $c$ son:

\n

$a=$[[0]]

\n

$b=$[[1]]

\n

$c=$[[2]]

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Para determinar la concavidad de la parábola debemos analizar el valor de $a$ en la función $f(x)=ax^2+bx+c$.

\n

Recordando la siguiente regla:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Valor de $a$ConvidadAbertura
\n

$a>0$

\n

Positivo

\n
Hacia arriba
\n

$a<0$

\n

Negativo

\n
Hacia abajo
\n

Por tanto, en la función $f(x)=x^2+6x+8$, el valor de $a=$[[0]], cuyo valor es [[1]] $0$, por tanto la parábola es [[2]].

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Para determinar el corte en el eje Y, debemos encontrar el valor de $c$ en la función $f(x)=x^2+6x+8$, obteniendo el punto del eje Y, cuyas coordenadas son $(0,c)=(0,$[[0]]$)$.

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Para determinar el eje de Simetría, debemos seguir los siguientes pasos:

\n

Paso 1: Primero debemos determinar los valores de $a$ y $b$ en la función $f(x)=x^2+6x+8$, los cuales son:

\n

$a=$[[0]]

\n

$b=$[[1]]

\n

Paso 2: Reemplazar $a$ y $b$ en la expresión $x=$$\\Large -\\frac{b}{2a}$, obteniendo que el eje de simetría es:

\n

                   [[1]]                [[1]]

\n

$x=-$___________$=-$__________$=$[[3]]

\n

                $2 \\cdot$ [[0]]              [[2]]

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Coordenada $x$:

\n

Recordemos que $x$ se obtiene de la expresión $x=$$\\Large -\\frac{b}{2a}$, valor que obtuvimos en el eje de simetría el cual es $x=$[[0]].

\n

Coordenada $y$:

\n

La coordenada $y$ se obtiene al evaluar $x=$[[0]] en la función $f(x)=x^2+6x+8$, obteniendo:

\n

$y=f($[[0]]$)=($[[0]]$)^2+6 \\cdot$[[0]]$+8$    

\n

$y=f($[[0]]$)=$[[1]][[2]]$+8$

\n

$y=f($[[0]]$)=$[[3]]

\n

Luego el vértice tiene por coordenadas $(x,y)=($[[4]]$,$[[5]]$)$.

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Para determinar los puntos de corte en el eje X debemos resolver la ecuación $f(x)=x^2+6x+8=0$, para ello usaremos la técnica de factorización de trinomios con término común.

\n

Paso 1: Debemos pensar en dos números $a$ y $b$ que multiplicados sean igual a $8$ y sumados sean igual a $2$ (considere $a$ mayor que $b$ para obtener la respuesta correcta)

\n

$a=$[[0]]

\n

$b=$[[1]]

\n

Paso 2: De lo anterior obtenemos la factorización y ecuación:

\n

$(x+a)\\cdot(x+b)=0$

\n

$(x+$ [[2]] $) \\cdot (x+$ [[3]] $)=0$

\n

Paso 3: Debemos resolver la ecuaciones de primer grado:

\n

$x+$[[2]]$=0$     y     $x+$[[3]]$=0$ 

\n

$x_1=$[[4]]       y     $x_2=$[[5]] 

\n

Paso 4: Determinar las coordenadas de corte del eje X, las cuales son $(x_1,0)=($[[6]]$,0)$ y $(x_2,0)=($[[7]]$,0)$.

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De acuerdo a lo anteriormente calculado determine cual de las siguientes gráficas corresponde a la gráfica de $f(x)=x^2+6x+8$.

\n

[[0]]

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