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In this section, we dive into the formal definition of a parabola and explore its properties. Learn how to identify the vertex, axis of symmetry, and the opening direction of a parabola equation.
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§10.4 The Parabola; Identifying Conic Sections
Equation of a Parabola We looked at parabolas in Chapter 8, viewing them as graphs of quadratic functions. In this section, we return again to the study of parabolas. Within this chapter on Conics, we extend our study by considering the formal definition of the parabola. A parabola is the set of all points in the plane that are equidistant from a fixed line (called the directrix) and a fixed point (called the focus). Note that the definition of both the hyperbola and the ellipse involved two fixed points, the foci. By contrast, the definition of a parabola is based on one point and a line. Blitzer, Algebra for College Students, 6e – Slide #2 Section 10.4
Equation of a Parabola Axis of Symmetry Focus Parabola Vertex Directrix Blitzer, Algebra for College Students, 6e – Slide #3 Section 10.4
Equation of a Parabola Blitzer, Algebra for College Students, 6e – Slide #4 Section 10.4
Equation of a Parabola CONTINUED y y (h, k) y = k y = k (h, k) x x Blitzer, Algebra for College Students, 6e – Slide #5 Section 10.4
Equation of a Parabola EXAMPLE Graph: SOLUTION We can graph this equation by following the steps in the preceding box. We begin by identifying values for a, k, and h. 1) Determine how the parabola opens. Note that a, the coefficient of , is 1. Thus, a > 0; this positive value tells us that the parabola opens to the right. Blitzer, Algebra for College Students, 6e – Slide #6 Section 10.4
Equation of a Parabola Blitzer, Algebra for College Students, 6e – Slide #7 Section 10.4
Equation of a Parabola CONTINUED 2) Find the vertex. The vertex of the parabola is at (h, k). Because k = 2 and h = 1, the parabola has its vertex at (1, 2). 3) Find the x-intercept. Replace y with 0 in The x-intercept is 5. The parabola passes through (5, 0). 4) Find the y-intercepts. Replace x with 0 in the given equation. This is the given equation. Replace x with 0. Subtract 1 from both sides. This equation clearly has no solutions since the left side is a negative number. Therefore, there are noy-intercepts. Blitzer, Algebra for College Students, 6e – Slide #8 Section 10.4
Equation of a Parabola CONTINUED 5) Graph the parabola. With a vertex at (1, 2), an x-intercept at 5, and no y-intercepts, the graph of the parabola is shown as follows. The axis of symmetry is the horizontal line whose equation is y = 2. Axis of symmetry: y = 2. Vertex: (1, 2) x-intercept: (5, 0) Blitzer, Algebra for College Students, 6e – Slide #9 Section 10.4
Equation of a Parabola EXAMPLE Graph: SOLUTION 1) Determine how the parabola opens. Note that a, the coefficient of , is -2. Thus a < 0; this negative value tells us that the parabola opens to the left. 2) Find the vertex. We know that the y-coordinate of the vertex is . We identify a, b, and c in a = -2 b = 4 c = -3 Blitzer, Algebra for College Students, 6e – Slide #10 Section 10.4
Equation of a Parabola CONTINUED Substitute the values of a and b into the equation for the y-coordinate: The y-coordinate of the vertex is 1. We substitute 1 for y into the parabola’s equation, to find the x-coordinate: The vertex is at (-1, 1). 3) Find the x-intercept. Replace y with 0 in The x-intercept is -3. The parabola passes through (-3, 0). Blitzer, Algebra for College Students, 6e – Slide #11 Section 10.4
Equation of a Parabola CONTINUED 4) Find the y-intercepts. Replace x with 0 in the given equation. This is the given equation. Replace x with 0. Use the quadratic formula to solve for y. Simplify. Subtract. This equation clearly has no solutions since the radicand is a negative number. Therefore, there are noy-intercepts. Blitzer, Algebra for College Students, 6e – Slide #12 Section 10.4
Equation of a Parabola CONTINUED 5) Graph the parabola. With a vertex at (-1, 1), an x-intercept at -3, and no y-intercepts, the graph of the parabola is shown below. The axis of symmetry is the horizontal line whose equation is y = 1. Axis of symmetry: y = 1. Vertex: (-1, 1) x-intercept: (-3, 0) Blitzer, Algebra for College Students, 6e – Slide #13 Section 10.4
Equations of Conic Sections Blitzer, Algebra for College Students, 6e – Slide #14 Section 10.4
Equations of Conic Sections EXAMPLE Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola: SOLUTION (Throughout the solution, in addition to identifying each equation’s graph, we’ll also discuss the graph’s important features.) If both variables are squared, the graph of the equation is not a parabola. In both cases, we collect the - and -terms on the same side of the equation. The graph cannot be a parabola. To see if it is a circle, an ellipse, or a hyperbola, we collect the - and -terms on the same side. Add to both sides. Blitzer, Algebra for College Students, 6e – Slide #15 Section 10.4
Equations of Conic Sections CONTINUED We obtain the following: Because the - and -terms have different coefficients of the same sign, the equation’s graph is an ellipse. The graph cannot be a parabola. To see if it is a circle, an ellipse, or a hyperbola, we collect the - and -terms on the same side. Subtract from both sides. We obtain: Because the - and -terms have coefficients with opposite signs, the equation’s graph is a hyperbola. Blitzer, Algebra for College Students, 6e – Slide #16 Section 10.4
The Parabola A parabola is the set of all points that are equidistant from a fixed line, the directrix, and a fixed point, the focus, that is not on the line. The line passing through the focus and perpendicular to the directrix is the axis of symmetry. The point of intersection of the parabola with its axis of symmetry is the vertex. Blitzer, Algebra for College Students, 6e – Slide #17 Section 10.4
