11.2 Parabolas

Graphing and Interpreting Parabolas

A parabola is a U-shaped curve defined by a squared term in its equation. In the context of conics, parabolas can open vertically or horizontally, and knowing how to tell the difference from an equation is a core skill for this unit.

Graphing Vertical and Horizontal Parabolas

Vertical parabolas open upward or downward. Their axis of symmetry is a vertical line.

Standard form: $y = a(x - h)^2 + k$

• $(h, k)$ is the vertex (the turning point of the curve).
• $a$ controls two things: direction and width.
  • $a > 0$: opens upward (U-shape)
  • $a < 0$: opens downward (inverted U)
  • $|a| > 1$: the parabola is narrower than $y = x^2$
  • $|a| < 1$: the parabola is wider than $y = x^2$

For example, $y = 3(x - 1)^2 + 2$ has its vertex at $(1, 2)$, opens upward, and is narrower than the basic parabola because $|a| = 3 > 1$.

Horizontal parabolas open to the left or right. Their axis of symmetry is a horizontal line.

Standard form: $x = a(y - k)^2 + h$

Notice the roles are swapped compared to the vertical form: the squared variable is now $y$, and the isolated variable is $x$.

• $(h, k)$ is still the vertex. Be careful here: $h$ is paired with $x$ and $k$ is paired with $y$, so in $x = 2(y - 3)^2 + 4$, the vertex is $(4, 3)$, not $(3, 4)$.
• $a$ works the same way for direction and width:
  • $a > 0$: opens to the right
  • $a < 0$: opens to the left
  • $|a| > 1$: narrower; $|a| < 1$: wider

Horizontal parabolas are not functions (they fail the vertical line test), but they're still important in the study of conics.

Key Features of Parabolas

Vertex: The point where the parabola changes direction. For vertical parabolas, this is the minimum (opens up) or maximum (opens down) point. Read it directly from the standard form as $(h, k)$.

Axis of symmetry: The line that splits the parabola into two mirror-image halves, passing through the vertex.

• Vertical parabolas: $x = h$
• Horizontal parabolas: $y = k$

Direction of opening: Determined entirely by the sign of $a$. A quick check of whether $a$ is positive or negative tells you which way the curve faces.

Intercepts: For vertical parabolas, the x-intercepts (if they exist) are where $y = 0$. Set the equation equal to zero and solve. A vertical parabola can have 0, 1, or 2 x-intercepts depending on whether the vertex sits above, on, or below the x-axis (for an upward-opening parabola; reverse for downward). The y-intercept is found by plugging in $x = 0$. For horizontal parabolas, the roles of x- and y-intercepts swap.

Steps to graph a parabola from standard form:

1. Identify whether the equation is vertical ($y = ...$ with $(x - h)^2$) or horizontal ($x = ...$ with $(y - k)^2$).

2. Read off the vertex $(h, k)$ and plot it.

3. Determine the direction of opening from the sign of $a$.

4. Draw the axis of symmetry through the vertex.

5. Choose a couple of convenient values on either side of the vertex, plug them in, and plot the resulting points.

6. Use symmetry to reflect those points across the axis and sketch the curve.

Applications of Parabolic Functions

Parabolas show up in real-world settings because of their reflective and optimizing properties.

Projectile motion traces a vertical, downward-opening parabola. The general height equation is:

$y = -\frac{1}{2}gt^2 + v_0 t + h_0$

where $g$ is gravitational acceleration ($9.8 \text{ m/s}^2$), $v_0$ is initial velocity, and $h_0$ is initial height. The vertex of this parabola gives you the maximum height and the time at which it occurs.

Reflective properties: Satellite dishes, headlights, and flashlights all use parabolic shapes. Any signal or light arriving parallel to the axis of symmetry reflects off the parabolic surface and converges at a single point called the focus. This is why dishes can collect weak signals and headlights can project a strong beam.

Optimization problems: In business contexts, profit as a function of price often forms a downward-opening parabola. The vertex represents the price that maximizes profit.

Parabolas as Conic Sections

A parabola is one of the four conic sections, the family of curves formed by slicing a double cone with a plane. The others are circles, ellipses, and hyperbolas. A parabola results when the cutting plane is parallel to the slant of the cone. This geometric origin is why parabolas appear alongside the other conics in this unit.