Question

eoc style question
the graph of the quadratic function $y = -x^2 - 2x + 3$ is shown below.
which statements about the graph are true? select all that apply.
□ the graph has roots at $-3$ and $1$.
□ the axis of symmetry is $x = -1$.
□ the graph has an $x$-intercept at $(1,0)$.
□ the graph has an $y$-intercept at $(-3,0)$.
□ the graph has a vertex at $(-1,4)$.

Explanation:

Step1: Encontrar las raíces (raíces)

Igualar $y=0$:
$$-x^2 -2x +3 = 0$$
Multiplicar por $-1$:
$$x^2 +2x -3 = 0$$
Factorizar:
$$(x+3)(x-1)=0$$
Resolver para $x$:
$x=-3$ o $x=1$

Step2: Calcular el eje de simetría

Usar la fórmula $x=-\frac{b}{2a}$ para $y=ax^2+bx+c$, donde $a=-1$, $b=-2$:
$$x=-\frac{-2}{2(-1)} = -1$$

Step3: Encontrar el vértice

Sustituir $x=-1$ en la función:
$$y=-(-1)^2 -2(-1)+3 = -1+2+3=4$$
El vértice es $(-1,4)$

Step4: Encontrar el intercepto en y

Sustituir $x=0$ en la función:
$$y=-(0)^2 -2(0)+3=3$$
El intercepto en y es $(0,3)$

Step5: Verificar interceptos en x

Las raíces son los interceptos en x: $(-3,0)$ y $(1,0)$

Answer:

• The graph has roots at $-3$ and $1$.
• The axis of symmetry is $x = -1$.
• The graph has an $x$-intercept at $(1,0)$.
• The graph has a vertex at $(-1,4)$.