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Chapter 4: Problem 106

a. Find the coordinates of the vertex of the parabola$$y=a x^{2}+b x+c, a \neq 0$$. b. When is the parabola concave up? Concave down? Give reasons for your answers.

Short Answer

Expert verified
The vertex is \((-\frac{b}{2a}, y_\text{vertex})\). The parabola is concave up if \(a > 0\) and concave down if \(a < 0\).

Step by step solution

01

Identify the standard form of the parabola

The given equation is in the form \( y = ax^2 + bx + c \), which is a quadratic equation representing a parabola.
02

Use the vertex formula

To find the vertex of the parabola, use the vertex formula. The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Substitute this value back into the equation \( y = ax^2 + bx + c \) to find the y-coordinate of the vertex.
03

Calculate the x-coordinate

Compute \( x_\text{vertex} = -\frac{b}{2a} \). This is the x-coordinate of the vertex of the parabola.
04

Calculate the y-coordinate

Substitute \( x_\text{vertex} \) into the equation \( y = ax^2 + bx + c \) to find \( y_\text{vertex} \). The vertex is \( (x_\text{vertex}, y_\text{vertex}) \).
05

Determine concavity condition (Concave Up)

If \( a > 0 \), the parabola opens upwards, and thus it is concave up.
06

Determine concavity condition (Concave Down)

If \( a < 0 \), the parabola opens downwards, and thus it is concave down.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex of a Parabola
To locate the vertex of a parabola, it's crucial to understand that it is the point where the parabola changes direction. For a parabola defined by the quadratic equation \( y = ax^2 + bx + c \), the vertex formula allows us to easily find its coordinates. The x-coordinate of the vertex can be found using: \[ x_{\text{vertex}} = -\frac{b}{2a} \]Once you have the x-coordinate, plug this value back into the original quadratic equation to find the y-coordinate, giving you the vertex coordinates \((x_{\text{vertex}}, y_{\text{vertex}})\). **Steps to Find the Vertex**:
  • Identify coefficients \(a\), \(b\), and \(c\) from the equation.
  • Compute \(x_{\text{vertex}} = -\frac{b}{2a}\).
  • Substitute \(x_{\text{vertex}}\) back into the equation to determine \(y_{\text{vertex}}\).
The vertex can sometimes represent the highest or lowest point on the graph, depending on the parabola's concavity.
Concavity
Concavity describes how a parabola curves. Whether a parabola is concave up or concave down is primarily determined by the coefficient \(a\) in the quadratic equation \(y = ax^2 + bx + c\). Concavity is a key concept in understanding the shape of a parabola.**Concave Up**:
  • If \(a > 0\), the parabola opens upwards.
  • This configuration resembles a smiling mouth, forming a "U" shape.
  • In this case, the vertex represents the lowest point, or the minimum of the parabola.
**Concave Down**:
  • If \(a < 0\), the parabola opens downwards.
  • This makes the parabola look like a frowning mouth, resembling an upside-down "U" shape.
  • Here, the vertex represents the highest point, or the maximum of the parabola.
Understanding concavity helps in predicting the general behavior of the quadratic function, which is vital in various applications of quadratics.
Quadratic Equations
Quadratic equations are polynomial equations of degree two, generally represented in standard form as \( y = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations form the basis for parabolas in coordinate geometry and appear frequently in mathematical analysis.**Features of Quadratic Equations**:
  • They have a set of solutions called roots, which can be real or complex.
  • The graph of a quadratic equation is a parabola, providing visual insights into the equation's behavior.
  • The sign of \(a\) determines the concavity of the parabola.
  • They can be solved using various methods including factoring, completing the square, or using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Quadratics are not just abstract mathematical concepts. They model many real-life phenomena, from the trajectory of objects in physics to profit calculations in economics. Thus, strengthening your grasp of quadratic equations is a valuable skill.

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