Problem 27 - Algebra 1

Question

Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry. $$ y=-16 x^{2} $$

Step-by-Step Solution

Answer

The graph of the function opens downwards, the coordinates of the vertex are (0,0), and the equation of the axis of symmetry is \( x = 0 \)

Step 1: Determine the Direction of Opening of the Parabola

A parabola opens upwards if its leading coefficient ( 'a' value ) is positive. It opens downwards if 'a' is negative. In the function \( y=-16x^2 \), 'a' is negative (-16). So, the parabola opens downwards.

Step 2: Find the Vertex Coordinates

In a standard form equation, the vertex of parabola \( f(x) = ax^2 + bx + c \) is given by \( (-b/2a , f(-b/2a) ) \). Here, 'a' and 'b' are coefficients of the quadratic term, and the linear term respectively. For our function \( f(x) = -16x^2 \), 'a' = -16 and 'b' = 0. Replacing these values in to the vertex equation, we find that the vertex is at point (0,0).

Step 3: Equation of the axis of symmetry

The axis of symmetry of a parabola \( y = ax^2 + bx + c \) is a vertical line given by the equation \( x = -b/2a \). For our function \( y=-16x^2 \), 'a' = -16 and 'b' = 0. So, the equation of the axis of symmetry is \( x = 0 \).

Key Concepts

Axis of Symmetry

Imagine folding a parabola along a straight line so that one side mirrors the other perfectly. This line is known as the axis of symmetry.
For any quadratic function in the form of \( y = ax^2 + bx + c \), the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \). It’s a vertical line that passes through the vertex of the parabola, effectively dividing it into two symmetrical halves.
Considering our example \( y = -16x^2 \), since the 'b' value (the coefficient of the x term) is zero, the equation of the axis of symmetry simplifies to \( x = 0 \). This means the axis of symmetry is the y-axis itself. Understanding where this axis lies is crucial for graphing the parabola and for predicting the parabola's behavior.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, often referred to as the 'peak'.
The position of the vertex can reveal a lot about the parabola's shape and orientation. For a quadratic function expressed as \( y = ax^2 + bx + c \), the vertex coordinates are \( (-\frac{b}{2a}, f(-\frac{b}{2a})) \).
In the case of the function \( y=-16x^2 \), we discover that the vertex is at (0, 0). This is because the 'b' coefficient is 0, making the calculation straightforward. The vertex at the origin indicates that this particular parabola is symmetrical about the y-axis and opens downward because the 'a' value is negative.

Direction of Parabola Opening

The direction of a parabola's opening is determined by the sign of the coefficient 'a' in the quadratic equation \( y = ax^2 + bx + c \).
If 'a' is positive, the parabola opens upwards, resembling a 'U' shape; if 'a' is negative, it opens downwards, like an upside-down 'U'.
For our given function \( y = -16x^2 \), the leading coefficient 'a' is -16, which is less than zero. Hence, the parabola opens downwards. This attribute affects many aspects, including the range of the function and the maximum or minimum value it can achieve, which in this instance is a maximum at the vertex (0,0). Grasping the direction is essential for sketching the graph and for applications involving projectile motion, where the path of an object thrown into the air describes a parabolic trajectory.