Problem 75 - Algebra 2

Question

Each point lies on a parabola with vertex \((0,2) .\) Write the equation of the parabola. $$ (1,-3) $$

Step-by-Step Solution

Answer

The equation of the parabola that has vertex at \((0,2)\) and goes through the point \((1,-3)\) is \(y = -5x^2 + 2\).

Step 1: Translate the Vertex

Switch the equation of the parabola, which is \(y = ax^2\), to account for the translated vertex \((0,2)\). The new equation becomes \(y-2 = ax^2\).

Step 2: Substitute the Given Point

Substitute the coordinates of the given point \((1, -3)\) into the equation to get an equation for the coefficient \(a\). The equation will now be \(-3-2 = a(1)^2\). This simplifies to \(-5 = a\).

Step 3: Final Equation

Now replace \(a\) with \(-5\) in the equation from Step 1 to get the final equation for the required parabola. Therefore, the final equation becomes \(y - 2 = -5x^2\). When rearranged, the equation becomes \(y = -5x^2 + 2\).

Key Concepts

Vertex Form

The vertex form of a parabolic equation is especially useful when you know the vertex of the parabola. This form is expressed as:

• \( y = a(x-h)^2 + k \)

In this expression:

• \((h, k)\) represents the vertex of the parabola.
• \(a\) is a coefficient that affects the width and direction of the parabola.

For instance, if the vertex is at \((0, 2)\), the equation simplifies to:

• \( y = a(x-0)^2 + 2 \)
• Or simply, \( y = ax^2 + 2 \).

This way, if you have a vertex and a point, you can find the specific equation of the parabola in vertex form.

Coordinate Geometry

Coordinate geometry is the study of geometric figures using a coordinate system. It bridges algebra with geometry, allowing us to solve geometric problems through equations. Below are a few key points about coordinate geometry:

• It uses labeled axes to plot points, lines, and curves.
• Distances and angles can be computed using algebraic formulas and equations.
• The position of points is given as ordered pairs \((x, y)\).
• Transformations like translations are used to position figures like parabolas.

Using coordinate geometry, we can easily find equations of lines, circles, and parabolas. For example, translating the vertex of a parabola doesn't change its shape, just its position in the coordinate plane. This technique is useful when solving problems involving parabolas, like finding its equation given a vertex and an additional point.

Quadratic Functions

Quadratic functions are a fundamental concept in algebra that help us understand parabolic shapes. A standard quadratic function is represented as:

• \( y = ax^2 + bx + c \).

Here's what you need to know:

• The graph of a quadratic function is a parabola.
• The direction of the parabola (upward or downward) is determined by the sign of \(a\):
  • If \(a > 0\), the parabola opens upward.
  • If \(a < 0\), it opens downward.
• The vertex of the parabola is its highest or lowest point.
• The coefficient \(a\) affects how "wide" or "narrow" the parabola is.

In the given exercise, we found \(a = -5\), meaning the parabola opens downward. The function does not have a linear term \(bx\), since the axis of symmetry is vertical and passes through the vertex given by the coordinate geometry.