Step 1: Identify the coefficients and constants in the quadratic function [tex]\( h(x) = 0.5(x + 2)^2 - 4 \)[/tex].
In the given quadratic function, the standard form is [tex]\( a(x - h)^2 + k \)[/tex].
We can compare the given function [tex]\( h(x) = 0.5(x + 2)^2 - 4 \)[/tex] with the standard form [tex]\( a(x - h)^2 + k \)[/tex]:
- Here, [tex]\( a \)[/tex] represents the coefficient of the squared term.
- The term inside the parentheses, [tex]\((x + 2)\)[/tex], can be rewritten as [tex]\((x - (-2))\)[/tex], hence, [tex]\( h = -2 \)[/tex].
- The constant term outside the squared term is [tex]\( k = -4 \)[/tex].
Thus, we find:
[tex]\[ a = 0.5 \][/tex]
[tex]\[ h = -2 \][/tex]
[tex]\[ k = -4 \][/tex]
So, the values are:
[tex]\[ a = 0.5 \][/tex]
[tex]\[ h = -2 \][/tex]
[tex]\[ k = -4 \][/tex]
Step 2: Describe the vertex of the parabola.
The vertex of the parabola in the standard form [tex]\( a(x - h)^2 + k \)[/tex] is given by the point [tex]\((h, k)\)[/tex].
Given the values we found:
[tex]\[ h = -2 \][/tex]
[tex]\[ k = -4 \][/tex]
Therefore, the vertex of the parabola is:
[tex]\[ (-2, -4) \][/tex]
Step 3: Determine the direction in which the parabola opens.
The direction in which the parabola opens is determined by the value of [tex]\( a \)[/tex]:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
Given:
[tex]\[ a = 0.5 \][/tex] (and [tex]\( 0.5 > 0 \)[/tex])
Since [tex]\( a \)[/tex] is positive, the parabola opens upwards.
Step 4: Identify the axis of symmetry.
The axis of symmetry for the parabola is the vertical line that passes through the vertex. It is given by the equation:
[tex]\[ x = h \][/tex]
Given [tex]\( h = -2 \)[/tex], the axis of symmetry is:
[tex]\[ x = -2 \][/tex]
Step 5: Plot key points and draw the graph.
To plot the graph, we plot the vertex [tex]\((-2, -4)\)[/tex] and find additional points on either side of the vertex.
For instance, we can choose a couple of [tex]\( x \)[/tex]-values near the vertex:
- If [tex]\( x = -3 \)[/tex]:
[tex]\[ h(x) = 0.5(-3 + 2)^2 - 4 = 0.5(-1)^2 - 4 = 0.5(1) - 4 = 0.5 - 4 = -3.5 \][/tex]
So, the point [tex]\((-3, -3.5)\)[/tex] is on the graph.
- If [tex]\( x = -1 \)[/tex]:
[tex]\[ h(x) = 0.5(-1 + 2)^2 - 4 = 0.5(1)^2 - 4 = 0.5(1) - 4 = 0.5 - 4 = -3.5 \][/tex]
So, the point [tex]\((-1, -3.5)\)[/tex] is on the graph.
We can plot more points similarly and then draw a smooth curve through these points to complete the parabola.
In summary:
- The vertex is [tex]\((-2, -4)\)[/tex].
- The parabola opens upwards.
- The axis of symmetry is [tex]\( x = -2 \)[/tex].
- Additional points such as [tex]\((-3, -3.5)\)[/tex] and [tex]\((-1, -3.5)\)[/tex] can be plotted.
By plotting these points and drawing a smooth curve through them, we can graph [tex]\( h(x) = 0.5(x + 2)^2 - 4 \)[/tex].
