Parabolas: Videos & Practice Problems

Understanding parabolas involves recognizing their standard forms: vertical parabolas follow $y=a(x-h{)}^2+k$ with vertex $(h,k)$ and axis of symmetry $x=h$. Horizontal parabolas switch variables and constants: $x=a(y-k{)}^2+h$, opening right if $a$ is positive, left if negative, with axis $y=k$. The focus and directrix define points equidistant from the parabola, essential in conic sections. Recognizing these forms aids in graphing and understanding polynomial behavior.

Parabolas In Standard Form

Quadratic functions produce graphs known as parabolas, which are U-shaped curves familiar in algebra and geometry. Typically, a quadratic equation in the form y = a(x − h)² + k represents a vertical parabola that opens either upward or downward. The vertex of this parabola is located at the point (h, k), and the axis of symmetry is the vertical line x = h. The coefficient a determines the direction of the opening: if a is positive, the parabola opens upward; if negative, it opens downward.

For example, consider a quadratic with a negative a value and vertex at (1, 3). This parabola opens downward with an axis of symmetry at x = 1. Understanding the vertex and axis of symmetry helps in graphing and analyzing the parabola’s shape and position.

Parabolas can also open horizontally, either to the left or right. This occurs when the roles of x and y are switched in the equation, resulting in the form x = a(y − k)² + h. Here, the vertex is at (h, k), but the axis of symmetry is horizontal, given by the line y = k. The sign of a again determines the direction: a positive a means the parabola opens to the right, while a negative a means it opens to the left.

For instance, if the equation is x = (y + 1)² − 2, rewriting it as x = 1(y − (−1))² + (−2) shows a positive a value of 1, indicating the parabola opens to the right. The vertex is at (−2, −1), and the axis of symmetry is the horizontal line y = −1.

Both vertical and horizontal parabolas are examples of conic sections, defined as the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. This geometric property means that for any point on the parabola, the distance to the focus equals the distance to the directrix. The directrix always lies outside the parabola, while the focus is located inside the curve, in the direction the parabola opens.

Recognizing the forms of parabolas and understanding their key features—vertex, axis of symmetry, direction of opening, focus, and directrix—are essential for graphing and analyzing quadratic functions and conic sections. These concepts provide a foundation for exploring more complex properties and applications of parabolas in mathematics and physics.

Find the vertex and axis of symmetry and determine the direction that the parabola opens.
$x=6y^2$

• A. Parabola opens to the right; Vertex: $(0,0)$; Axis of Symmetry: $x=0$.
• B. Parabola opens to the left; Vertex: $(0,6)$; Axis of Symmetry: $x=0$.
• C. Parabola opens to the right; Vertex: $\left(0,0\right)$; Axis of Symmetry: $y=0$.
• D. Parabola opens to the left; Vertex: $\left(0,0\right)$; Axis of Symmetry: $y=0$.

Answer: (C)

Find the vertex and axis of symmetry and determine the direction that the parabola opens.
$y={(x+4)}^2-9$

• A. Parabola opens downwards; Vertex: $(-4,-9)$; Axis of symmetry: $x=-4$
• B. Parabola opens upwards; Vertex: $\left(-4,-9\right)$; Axis of symmetry: $x=-4$
• C. Parabola opens upwards; Vertex: $\left(4,-9\right)$; Axis of symmetry: $x=4$
• D. Parabola opens downwards; Vertex: $\left(4,-9\right)$; Axis of symmetry: $x=4$

Answer: (B)

Find the vertex and axis of symmetry and determine the direction that the parabola opens.
$x=-2y^2+6$

• A. Parabola opens to the left; Vertex: $(0,6)$; Axis of Symmetry: $y=0$
• B. Parabola opens to the right; Vertex: $(6,0)$; Axis of Symmetry: $y=2$
• C. Parabola opens to the left; Vertex: $\left(6,0\right)$; Axis of Symmetry: $y=0$
• D. Parabola opens to the right; Vertex: $\left(0,6\right)$; Axis of Symmetry: $y=2$

Answer: (C)

Here’s what students ask on this topic:

What is the standard form equation of a vertical parabola and how do you determine its vertex and axis of symmetry?

The standard form equation of a vertical parabola is given by $y=a(x-h){²}^{}+k$, where $a$ determines the direction the parabola opens. If $a$ is positive, the parabola opens upward; if negative, it opens downward. The vertex of the parabola is the point $(h,k)$, which represents the maximum or minimum point on the graph. The axis of symmetry is a vertical line that passes through the vertex, given by $x=h$. This axis divides the parabola into two mirror-image halves. Understanding these components helps in graphing and analyzing the parabola's properties effectively.

How do you write the equation of a horizontal parabola and identify its vertex and axis of symmetry?

A horizontal parabola's equation is written as $x=a\left(y-k\right){²}^{}+h$. This form is derived by switching the roles of $x$ and $y$ from the vertical parabola equation. The vertex is located at $(h,k)$. The value of $a$ determines the direction the parabola opens: if $a$ is positive, it opens to the right; if negative, it opens to the left. The axis of symmetry for a horizontal parabola is a horizontal line given by $y=k$. This axis divides the parabola into two symmetric parts horizontally. Recognizing these features is essential for graphing and understanding horizontal parabolas.

What is the significance of the focus and directrix in the definition of a parabola?

The focus and directrix are fundamental in defining a parabola as a conic section. A parabola consists of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix. The focus lies inside the parabola, in the direction it opens, while the directrix is a line outside the parabola. This geometric property ensures that for any point on the parabola, the distance to the focus equals the perpendicular distance to the directrix. Understanding the focus and directrix helps in comprehending the parabola's shape and its reflective properties, which are important in applications like satellite dishes and headlights.

How does the sign of the coefficient 'a' affect the direction in which a parabola opens?

The coefficient $a$ in the parabola's equation determines the direction the parabola opens. For a vertical parabola with equation $y=a\left(x-h\right){²}^{}+k$, if $a$ is positive, the parabola opens upward; if negative, it opens downward. For a horizontal parabola with equation $x=a\left(y-k\right){²}^{}+h$, a positive $a$ means the parabola opens to the right, while a negative $a$ means it opens to the left. This sign is crucial for graphing and understanding the parabola's orientation.

How do you find the vertex of a parabola given its equation in standard form?

To find the vertex of a parabola given in standard form, identify the values of $h$ and $k$ from the equation. For a vertical parabola, the standard form is $y=a\left(x-h\right){²}^{}+k$, and the vertex is at the point $(h,k)$. For a horizontal parabola, the equation is $x=a\left(y-k\right){²}^{}+h$, and the vertex is also at $(h,k)$. The vertex represents the parabola's highest or lowest point (vertical) or the leftmost or rightmost point (horizontal), depending on the parabola's orientation.