Parabolas: Videos & Practice Problems

Parabolas can be derived from slicing a cone with a tilted plane, resulting in a focus point and a directrix line. The focus is a point on the graph, while the directrix is a line, both equidistant from the vertex. The equation for a vertical parabola is $(y-k)=4p(x-h)$. For horizontal parabolas, the equation switches to $(x-h)=4p(y-k)$. The sign of p determines the direction of the opening.

Parabolas as Conic Sections

In the study of conic sections, the parabola is a significant shape that can be visualized by slicing a three-dimensional cone with a tilted plane. Understanding parabolas involves identifying two key components: the focus and the directrix. The focus is a specific point on the graph, while the directrix is a line. Both the focus and the directrix are equidistant from the vertex of the parabola, although they may be oriented in different directions.

When analyzing a parabola, the orientation determines the location of the focus and directrix. For a parabola that opens upwards, the focus is positioned above the vertex, while the directrix is below it. Conversely, if the parabola opens downwards, the focus is below the vertex, and the directrix is above it. The relationship between the vertex, focus, and directrix is crucial for graphing parabolas accurately.

The standard form of a parabola's equation is given by:

$$y = \frac{1}{4p}(x - h)^2 + k$$

In this equation, \((h, k)\) represents the vertex, and \(p\) is a parameter that indicates the distance from the vertex to the focus and the directrix. A positive \(p\) value indicates that the parabola opens upwards, while a negative \(p\) value indicates it opens downwards.

To find the focus and directrix, one must first determine the value of \(p\). For example, if the equation of the parabola is given as \(y - 2 = \frac{1}{8}(x - 1)^2\), we can identify that \(4p = 8\). Solving for \(p\) gives:

$$p = \frac{8}{4} = 2$$

With \(p\) known, the focus can be located by moving \(p\) units from the vertex. If the vertex is at \((1, 2)\), the focus would be at \((1, 2 + 2) = (1, 4)\). The directrix, being \(p\) units in the opposite direction, would be at \(y = 2 - 2 = 0\).

To graph the parabola, one can also determine its width by moving \(2p\) units horizontally from the focus. In this case, \(2p = 4\), allowing us to find points at \((1 - 4, 4) = (-3, 4)\) and \((1 + 4, 4) = (5, 4)\). Connecting these points with a smooth curve forms the parabola.

In summary, understanding the focus, directrix, and the equation of a parabola is essential for graphing and solving problems related to conic sections. By following a systematic approach to identify these components, one can effectively analyze and represent parabolas in mathematical contexts.

Graph the parabola $-4\left(y+1\right)=\left(x+1\right)^2$, and find the focus point and directrix line.

• A.
• B.
• C.
• D.

Answer: (D)

If a parabola has the focus at $\left(0,-1\right)$ and a directrix line $y=1$, find the standard equation for the parabola.

• A. $4y=x^2$
• B. $4\left(y-1\right)=x^2$
• C. $-4y=x^2$
• D. $-4\left(y+1\right)=x^2$

Answer: (C)

Parabolas as Conic Sections Example 1

To analyze parabolas, we utilize the standard equation of a parabola, which is given by:

\( 4p(y - k) = (x - h)^2 \)

In this equation, \( (h, k) \) represents the vertex of the parabola. If \( h \) and \( k \) are absent, the parabola is centered at the origin, simplifying the equation to:

\( 4py = x^2 \)

When determining the vertex, focus, and directrix, we can follow a systematic approach. For example, consider a parabola represented by \( 4py = x^2 \). Here, the vertex is at the origin \( (0, 0) \). To find the focus and directrix, we set \( 4p \) equal to the coefficient in front of \( y \). If \( 4p = 16 \), solving for \( p \) gives:

\( p = \frac{16}{4} = 4 \)

Since \( p \) is positive, the parabola opens upward. The focus, therefore, is located at \( (0, 4) \), and the directrix is the line \( y = -4 \).

In another example, if we have a parabola in the form \( 4py = \frac{1}{3}x^2 \), we again identify the vertex at the origin. Setting \( 4p = \frac{1}{3} \) leads to:

\( p = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12} \)

With \( p \) being positive, the focus is at \( (0, \frac{1}{12}) \) and the directrix is at \( y = -\frac{1}{12} \).

For parabolas not centered at the origin, such as \( 4p(y - 1) = (x - 2)^2 \), we identify the vertex as \( (2, 1) \). Setting \( 4p = 8 \) gives:

\( p = \frac{8}{4} = 2 \)

Since \( p \) is positive, the focus is at \( (2, 3) \) and the directrix is the line \( y = -1 \).

Lastly, for a parabola like \( -12y = (x + 1)^2 \), we recognize the vertex is at \( (-1, 0) \). Setting \( 4p = -12 \) results in:

\( p = -3 \)

Here, \( p \) is negative, indicating the parabola opens downward. Thus, the focus is at \( (-1, -3) \) and the directrix is the line \( y = 3 \).

In summary, by identifying the vertex, calculating \( p \), and determining the focus and directrix, we can effectively analyze the properties of any parabola using its standard equation.

Horizontal Parabolas

In the study of parabolas, we typically encounter vertical parabolas that open either upwards or downwards. However, when we consider horizontal parabolas, the equations and characteristics change slightly, primarily involving the switching of the x and y variables. This transformation leads to a new understanding of how these parabolas behave and how to graph them effectively.

For horizontal parabolas, the general form of the equation is derived from the vertical parabola's equation by interchanging the roles of x and y. Specifically, the equation for a vertical parabola can be expressed as:

$$4py = x^2$$

For a horizontal parabola, this becomes:

$$4px = y^2$$

In these equations, the parameter p represents the distance from the vertex to the focus, and it also determines the direction in which the parabola opens. If p is positive, the horizontal parabola opens to the right; if p is negative, it opens to the left. Conversely, for vertical parabolas, a positive p indicates an upward opening, while a negative p indicates a downward opening.

When graphing a horizontal parabola, the first step is to identify the vertex, which is often at the origin (0,0) if no h or k values are present. Next, the p value can be calculated from the equation. For instance, if the equation is given as:

$$y^2 = 8x$$

we can set 4p equal to 8, leading to:

$$p = \frac{8}{4} = 2$$

With p determined, the focus can be located by moving p units in the direction the parabola opens. In this case, the focus would be at (2,0). The width of the parabola is found by moving up and down 2p units from the focus, resulting in points at (2,4) and (2,-4).

Finally, the directrix, which is always opposite the direction of the opening, can be established. For a parabola that opens to the right, the directrix is a vertical line located p units to the left of the vertex, represented by the equation:

$$x = -2$$

By connecting these points with a smooth curve, one can accurately graph the horizontal parabola. Understanding these principles allows for a comprehensive approach to analyzing and graphing both vertical and horizontal parabolas effectively.

Graph the parabola $8\left(x+1\right)=\left(y-2\right)^2$ , and find the focus point and directrix line.

• A.
• B.
• C.
• D.

Answer: (B)

If a parabola has the focus at $\left(2,4\right)$ and a directrix line $x=-4$ , find the standard equation for the parabola.

• A. $12\left(x+1\right)=\left(y-4\right)^2$
• B. $-\left(x+1\right)=\left(y-4\right)^2$
• C. $12x=y^2$
• D. $4\left(x-1\right)=\left(y+4\right)^2$

Answer: (A)

Horizontal Parabolas Example 1

In studying parabolas, it's essential to understand their geometric properties, particularly the vertex, focus, and directrix. A horizontal parabola can be represented by the equation:

\(4p(x - h) = (y - k)^2\)

Here, \((h, k)\) denotes the vertex, and \(p\) indicates the distance from the vertex to the focus and the directrix. A positive \(p\) value signifies that the parabola opens to the right, while a negative \(p\) indicates it opens to the left.

For example, consider a horizontal parabola where the equation is given as \(y^2 = 4x\). In this case, since there are no constants subtracted from \(x\) or \(y\), the vertex is at the origin \((0, 0)\). Setting \(4p = 4\) gives \(p = 1\), indicating the focus is at \((1, 0)\) and the directrix is the line \(x = -1\).

In another example, if the equation is \(y^2 = 9 - 4x\), we can rewrite it to identify the vertex. Here, \(h = 0\) and \(k = 2\) (since \(y - 2\) is equivalent to \(y + 2\)). The vertex is at \((0, 2)\). Setting \(4p = 9\) results in \(p = \frac{9}{4}\), meaning the focus is at \((\frac{9}{4}, 2)\) and the directrix is at \(x = -\frac{9}{4}\).

For a more complex case, consider the equation \( (y + 2)^2 = 16(x - 4) \). Here, \(h = 4\) and \(k = -2\), giving a vertex at \((4, -2)\). Setting \(4p = 16\) results in \(p = 4\), placing the focus at \((8, -2)\) and the directrix at \(x = 0\).

Lastly, if the equation is \( (y - 0)^2 = -2(x - 1) \), we find \(h = 1\) and \(k = 0\), so the vertex is at \((1, 0)\). Here, \(4p = -2\) leads to \(p = -\frac{1}{2}\), indicating the parabola opens to the left. The focus is at \((\frac{1}{2}, 0)\) and the directrix is the line \(x = \frac{3}{2}\).

Understanding these properties allows for the effective analysis of parabolas, enabling the identification of their key features based on their equations.

Here’s what students ask on this topic:

What is the standard form equation of a parabola?

The standard form equation of a parabola depends on its orientation. For a vertical parabola, the equation is $(y-k)=4p(x-h)$. For a horizontal parabola, the equation is $(x-h)=4p(y-k)$. Here, (h, k) is the vertex of the parabola, and p is the distance from the vertex to the focus or directrix.

How do you find the focus and directrix of a parabola?

To find the focus and directrix of a parabola, you need the value of p from the standard form equation. For a vertical parabola $(y-k)=4p(x-h)$, the focus is at (h, k + p) and the directrix is the line y = k - p. For a horizontal parabola $(x-h)=4p(y-k)$, the focus is at (h + p, k) and the directrix is the line x = h - p.

What is the difference between vertical and horizontal parabolas?

The main difference between vertical and horizontal parabolas lies in their orientation and equations. A vertical parabola opens either up or down and has the equation $(y-k)=4p(x-h)$. A horizontal parabola opens either left or right and has the equation $(x-h)=4p(y-k)$. The sign of p determines the direction of the opening: positive p means up or right, and negative p means down or left.

How do you graph a parabola given its equation?

To graph a parabola given its equation, follow these steps: 1) Identify the vertex (h, k) from the equation. 2) Calculate the value of p from the coefficient of the linear term. 3) Determine the focus and directrix using the value of p. 4) Plot the vertex, focus, and directrix on the graph. 5) Draw the parabola by connecting points that are equidistant from the focus and directrix. For vertical parabolas, use $(y-k)=4p(x-h)$, and for horizontal parabolas, use $(x-h)=4p(y-k)$.

What is the significance of the p value in a parabola's equation?

The p value in a parabola's equation is crucial as it determines the distance between the vertex and the focus, as well as the vertex and the directrix. For a vertical parabola $(y-k)=4p(x-h)$, a positive p value means the parabola opens upwards, while a negative p value means it opens downwards. For a horizontal parabola $(x-h)=4p(y-k)$, a positive p value means the parabola opens to the right, while a negative p value means it opens to the left.