Question

Sketch the three basic conic sections in standard position with vertices and foci on the \(x\) -axis.

Short answer

Question: Explain how to sketch the basic conic sections (parabola, ellipse, and hyperbola) in standard position with vertices and foci on the x-axis. Mention the standard equations and notable features for each conic section.

Step-by-step solution

Parabola

To sketch a parabola in standard position with vertex at the origin and the focus on the x-axis, its equation takes the form \(y^2 = 4px\), where p is the distance between the vertex and the focus. Because the parabola is symmetric with respect to the x-axis, we can draw a horizontal line representing the directrix on the other side of the vertex (to the left for p > 0 or to the right for p < 0).

Ellipse

To sketch an ellipse in standard position with its center at the origin and foci on the x-axis, the equation is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a > b \ge 0\). The foci are located at \((\pm c, 0)\), where \(c = \sqrt{a^2 - b^2}\), and the endpoints of the major and minor axes are \((\pm a, 0)\) and \((0, \pm b)\). We can draw the vertices, foci, and the ellipse with the given information.

Hyperbola

To sketch a hyperbola in standard position with its center at the origin, vertices on the x-axis, and opening along the x-axis, the equation is given by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a > 0\) and \(b > 0\). The vertices are located at \((\pm a, 0)\), and the foci at \((\pm c, 0)\), where \(c = \sqrt{a^2 + b^2}\). Additionally, we can draw the asymptotes by finding their slopes \(m = \pm\frac{b}{a}\) and using points from the vertices to calculate their equations, which are given by \(y = \pm\frac{b}{a}x\). Now, we can sketch the hyperbola, vertices, foci, and the asymptotes.

Key concepts

Parabola

A parabola is a simple yet intriguing conic section. When you imagine its shape, think about the path of a ball being thrown into the air and then coming down. Mathematically, a parabola opens either upwards or sideways, depending on the axis it is aligned with. For a parabola with a vertex located at the origin and a focus on the x-axis, the equation is given by \(y^2 = 4px\). Here, \(p\) represents the distance from the vertex to the focus. This form suggests that the parabola is symmetric about the x-axis.

### Key Features of a Parabola- **Vertex:** The starting point of the parabola at the origin \((0, 0)\).- **Focus:** A point on the x-axis \((p, 0)\) where all the parabola's paths focus.- **Directrix:** A line perpendicular to the axis of symmetry, located \(p\) units on the opposite side of the vertex as the focus.

By sketching, you can draw the axis of symmetry, mark the vertex, and maintain equal distances to the focus and directrix. This leads to a clear visual understanding of the parabola's shape.

Ellipse

An ellipse represents the set of all points where the sum of the distances from two focus points is constant. You might visualize it as a stretched circle, like a squashed balloon. For an ellipse centered at the origin with its major axis along the x-axis, use the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

### Components of an Ellipse- **Center:** The midpoint located at the origin \((0, 0)\).- **Vertices:** The endpoints of the major axis are \((a, 0)\) and \((-a, 0)\).- **Co-vertices:** The endpoints of the minor axis are \((0, b)\) and \((0, -b)\).- **Foci:** Positioned along the x-axis at \(( c, 0)\) and \((-c, 0)\), where \(c = \sqrt{a^2 - b^2}\).

This configuration shows that the ellipse will be wider than tall if \(a > b\). By plotting the vertices, co-vertices, and foci, you gain a firm geometric sense of the ellipse.

Hyperbola

A hyperbola seems like an exaggerated version of an ellipse, yet it behaves quite differently. The shape reveals two opposite sections, akin to two mirrored parabolas. In the standard form where the hyperbola is centered at the origin and opens along the x-axis, its equation can be expressed as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

### Understanding a Hyperbola- **Vertices:** Found on the x-axis at \((a, 0)\) and \((-a, 0)\).- **Foci:** These pivotal points are located at \((c, 0)\) and \((-c, 0)\), with \(c = \sqrt{a^2 + b^2}\).- **Asymptotes:** Lines that the hyperbola approaches but never touches, with slopes \(\pm\frac{b}{a}\). Their equations include \(y = \pm\frac{b}{a}x\).

By sketching these elements, especially the asymptotes, you demonstrate the hyperbola's nature and direction. Observing how it curves outward away from the center helps highlight the distinctive features of this intriguing conic section.