Problem 25 - 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. $$ y=-5 x^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Direction of the Parabola

The sign of \(a\) in the quadratic equation \(y = ax^{2}\) determines the direction of the parabola. If \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, the parabola opens downwards. In this case, \(a = -5\), which is negative. Therefore, the graph of the function opens downwards.

Step 2: Find the Vertex

For a quadratic function in the form \(y = ax^{2}\), the vertex is always at the origin, (0, 0). This is because the function is symmetric with respect to the y-axis.

Step 3: Equation of the Axis of Symmetry

The axis of symmetry for a parabola rooted at the origin is always the y-axis, which is the vertical line \(x = 0\).

Key Concepts

Parabola

A parabola is the U-shaped graph that is the visualization of a quadratic function. Understanding the shape of a parabola is essential to grasping how quadratic functions work. Here are some key things to remember:

• The parabola can either open upwards or downwards.
  
• Whether a parabola opens up or down depends on the sign of the coefficient \( a \) in the quadratic function \( y = ax^{2}\).
  
• If \( a \) is positive, the parabola opens upwards like a smile. If \( a \) is negative, it opens downwards like a frown.

For instance, in the quadratic function \( y = -5x^{2} \), the coefficient \( a \) is -5. Since it's negative, the graph of this quadratic function is a parabola that opens downwards. Recognizing this pattern helps in predicting the behavior and visual shape of quadratic functions.

Vertex

The vertex of a parabola is a special point where the direction of the curve changes. It is often thought of as the "tip" or the "turning point" of the parabola. Here's more on understanding the vertex:

• In the simplest form of a quadratic equation \( y = ax^{2} \), the vertex is at the origin, i.e., at point (0, 0).
  
• The vertex represents the maximum or minimum point of the parabola. In a downward-opening parabola, the vertex is the highest point on the graph.
  
• The position of the vertex can be found using the formula \( x = -\frac{b}{2a} \) for a general quadratic \( y = ax^{2} + bx + c \). However, if \( b \) is 0, the vertex is straightforwardly at the y-axis.

For our function \( y = -5x^{2} \), the vertex is at the origin, (0, 0), because it is symmetric with respect to the y-axis and does not include any \( bx \) or \( c \) terms.

Axis of Symmetry

The axis of symmetry is a critical element in understanding parabolas. It is the line that divides the parabola into two mirror-image halves. This line helps in identifying the parabola's symmetry and is key to finding other properties:

• For any quadratic function \( y = ax^{2} + bx + c \), the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \).
  
• In the case of parabolas that are straightforwardly represented as \( y = ax^{2} \), the axis of symmetry is simply the y-axis, given as \( x = 0 \).
• The axis of symmetry passes through the vertex of the parabola, ensuring both sides are symmetric.

In the quadratic function \( y = -5x^{2} \), since it is in its simplest form, the axis of symmetry is the vertical line \( x = 0 \), identical to the line of symmetry along the y-axis. Recognizing this helps to graph these functions more accurately and to foresee their symmetrical properties.