Problem 5 - Intermediate Algebra

Question

The graph of each equation is a parabola. Determine whether the parabola opens upward, downward, to the left, or to the right. Do not graph. $$y=-x^{2}+2 x+1$$

Step-by-Step Solution

Answer

The parabola opens downward.

Step 1: Identify the General Form

The equation given is in the form of a quadratic equation: \( y = ax^2 + bx + c \). Here, \( a = -1 \), \( b = 2 \), and \( c = 1 \).

Step 2: Determine the Direction of the Parabola

The direction of a parabola (upward or downward) when expressed in the form \( y = ax^2 + bx + c \) is determined by the coefficient \( a \). If \( a > 0 \), the parabola opens upward. If \( a < 0 \), it opens downward. Here, \( a = -1 \) which is less than 0, so the parabola opens downward.

Key Concepts

Parabola Direction

To determine the direction in which a parabola opens, we must first take a closer look at the quadratic equation in standard form, which is \( y = ax^2 + bx + c \). The key factor here is the coefficient \( a \). This coefficient plays a vital role in dictating the orientation of the parabola's opening.

• If \( a > 0 \), the parabola opens upwards. Imagine it as a U-shaped curve, with the arms reaching towards the sky.
• If \( a < 0 \), like in our equation where \( a = -1 \), the parabola opens downwards. Picture it as an upside-down U, with the curve aiming towards the ground.

It’s important to remember that for graphing purposes in this context, we are only concerned with whether the parabola opens upwards or downwards since the quadratic expression is in terms of \( y \). Quadratics expressed in this manner will never open left or right.

Quadratic Graphing

Graphing a quadratic equation can initially feel like a daunting task, but understanding the basic principles can simplify the process significantly. Quadratic graphs are always parabolic, meaning they form a symmetrical "U" shape. This shape can either open upwards or downwards, which is something we determine from the coefficient \( a \) in the equation \( y = ax^2 + bx + c \).
Additionally, identifying specific characteristics of a quadratic graph aids in graphing:

• Vertex: The highest or lowest point on the parabola, depending on its orientation. This is the point where the graph changes direction.
• Axis of Symmetry: A vertical line through the vertex around which the parabola is symmetrical.
• Y-Intercept: The point at which the parabola crosses the y-axis, determined by the constant \( c \) in the equation.

Graphing manually requires calculating several values of \( x \) to determine corresponding \( y \) values, plotting these points, and sketching the curve. However, with practice, understanding these elements will make graphing quicker and easier.

Quadratic Coefficients

The coefficients in a quadratic equation \( y = ax^2 + bx + c \) provide significant information about the parabola's shape and position. Each coefficient has a distinct role:

• Coefficient \( a \): Determines the direction and "width" of the parabola. A larger absolute value of \( a \) makes the parabola "narrower," while a smaller absolute value makes it "wider." For instance, in our example equation, \( a = -1 \), which dictates that the parabola opens downward.
• Coefficient \( b \): Impacts the position of the parabola along the x-axis and affects the symmetry axis's location. It can be said to "tilt" the parabola slightly to the left or right, although it does not affect the width or direction.
• Coefficient \( c \): Represents the y-intercept of the quadratic graph. This is where the parabola will intersect the y-axis \((x = 0)\). In our given equation, \( c = 1 \), meaning the parabola will touch the y-axis at this point.

Understanding these coefficients helps visualize the quadratic graph by indicating its opening direction, spacing, and exact placement on the coordinate system.