Question

graph the parabola.
( y = (x + 4)^2 + 3 )
plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. then click on the graph-a-function button.

Explanation:

Step1: Find the vertex

The equation of the parabola is in vertex form \( y = (x - h)^2 + k \), where \((h, k)\) is the vertex. For \( y=(x + 4)^2+3 \), we can rewrite it as \( y=(x - (-4))^2+3 \), so the vertex \((h, k)\) is \((-4, 3)\).

Step2: Find points to the left of the vertex

Let's choose \( x=-5 \) (left of \( x = - 4\)):
Substitute \( x=-5 \) into \( y=(x + 4)^2+3 \):
\( y=(-5 + 4)^2+3=(-1)^2+3=1 + 3 = 4 \). So the point is \((-5, 4)\).
Choose \( x=-6 \) (left of \( x=-4\)):
Substitute \( x = - 6\) into \( y=(x + 4)^2+3 \):
\( y=(-6 + 4)^2+3=(-2)^2+3=4 + 3 = 7 \). So the point is \((-6, 7)\).

Step3: Find points to the right of the vertex

Let's choose \( x=-3 \) (right of \( x=-4\)):
Substitute \( x=-3 \) into \( y=(x + 4)^2+3 \):
\( y=(-3 + 4)^2+3=(1)^2+3=1 + 3 = 4 \). So the point is \((-3, 4)\).
Choose \( x=-2 \) (right of \( x=-4\)):
Substitute \( x=-2 \) into \( y=(x + 4)^2+3 \):
\( y=(-2 + 4)^2+3=(2)^2+3=4 + 3 = 7 \). So the point is \((-2, 7)\).

Now we have the five points: vertex \((-4, 3)\), left points \((-5, 4)\), \((-6, 7)\) and right points \((-3, 4)\), \((-2, 7)\). We can plot these points on the graph.

Answer:

The vertex is \((-4, 3)\), left points are \((-5, 4)\), \((-6, 7)\) and right points are \((-3, 4)\), \((-2, 7)\) (these points can be plotted to graph the parabola).