Graph -1/2(x-2)^2-4
The question provided seems to be asking for a graphical representation of the quadratic function described by the equation $-1/2(x-2)^2-4$. This involves plotting the given equation on a coordinate plane, typically with the x-axis representing the independent variable and the y-axis representing the dependent variable. The equation outlines a parabola that has been transformed from the standard form of a quadratic equation. The form of the equation suggests that certain transformations have been applied to the basic $y = x^2$parabola, such as vertical stretching/compression, horizontal shifting, and vertical shifting. The specific transformations are indicated by the constants and expressions within the equation. The parabola will open either upwards or downwards and will be positioned on the graph according to the transformations dictated by the values in the equation.
$- \frac{1}{2} \left(\left(\right. x - 2 \left.\right)\right)^{2} - 4$
Answer
Solution:
Step:1
Analyze the characteristics of the parabola in question.
Step:1.1
Identify the parameters $a$, $h$, and $k$ from the vertex form equation $y = a(x - h)^{2} + k$.
$$a = -\frac{1}{2}, h = 2, k = -4$$
Step:1.2
The parabola opens downwards because $a$ is less than zero.
Step:1.3
Determine the vertex $(h, k)$.
$$(2, -4)$$
Step:1.4
Calculate the distance $p$ from the vertex to the focus of the parabola.
Step:1.4.1
Use the formula $p = \frac{1}{4a}$ to find $p$.
Step:1.4.2
Plug in the value of $a$.
$$p = \frac{1}{4(-\frac{1}{2})}$$
Step:1.4.3
Simplify the expression.
Step:1.4.3.1
Eliminate the common factor.
Step:1.4.3.1.1
Express $1$ as $-1(-1)$.
$$p = \frac{-1(-1)}{4(-\frac{1}{2})}$$
Step:1.4.3.1.2
Place the negative sign outside the fraction.
$$p = -\frac{1}{4(\frac{1}{2})}$$
Step:1.4.3.2
Combine the numbers $4$ and $\frac{1}{2}$.
$$p = -\frac{1}{2}$$
Step:1.5
Locate the focus of the parabola.
Step:1.5.1
To find the focus, add $p$ to the vertex's y-coordinate $k$ for parabolas that open up or down.
$$(h, k + p)$$
Step:1.5.2
Insert the values of $h$, $k$, and $p$ and simplify.
$$(2, -\frac{9}{2})$$
Step:1.6
The axis of symmetry is the line that passes through both the vertex and the focus.
$$x = 2$$
Step:1.7
Determine the directrix of the parabola.
Step:1.7.1
The directrix is a horizontal line calculated by subtracting $p$ from the vertex's y-coordinate $k$ for parabolas that open up or down.
$$y = k - p$$
Step:1.7.2
Substitute the values for $p$ and $k$ and simplify.
$$y = -\frac{7}{2}$$
Step:1.8
Graph the parabola using its properties.
Direction: Opens Down Vertex: $(2, -4)$ Focus: $(2, -\frac{9}{2})$ Axis of Symmetry: $x = 2$ Directrix: $y = -\frac{7}{2}$
Step:2
Choose several values for $x$ and calculate the corresponding $y$ values, focusing on points around the vertex.
Step:2.1 - Step:2.12
For each selected $x$ value, substitute it into the equation and simplify to find the $y$ value.
Step:2.13
Plot the points on the graph.
$$ \begin{array}{c|c} x & y \\ \hline 0 & -6 \\ 1 & -\frac{9}{2} \\ 2 & -4 \\ 3 & -\frac{9}{2} \\ 4 & -6 \\ \end{array} $$
Step:3
Complete the graph of the parabola using the identified properties and plotted points.
Knowledge Notes:
The vertex form of a parabola's equation is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola, and $a$ determines the direction and width of the parabola.
If $a$ is negative, the parabola opens downwards; if $a$ is positive, it opens upwards.
The focus of a parabola is a point from which distances to any point on the parabola are equidistant from the directrix, a line perpendicular to the axis of symmetry.
The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
The distance $p$ from the vertex to the focus (or directrix) is given by $p = \frac{1}{4a}$.
To graph a parabola, it's helpful to find the vertex, focus, axis of symmetry, and directrix. Plotting additional points by selecting $x$ values near the vertex and calculating the corresponding $y$ values can provide a more accurate graph.
When simplifying expressions, common mathematical operations such as factoring, finding common denominators, and combining like terms are used to reach the simplest form.
