Problem 22 - 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. $$ y=-7 x^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Determine the Direction of Opening

Since the coefficient of \(x^2\) is -7, which is less than 0, the graph of the function opens downward.

Step 2: Find the Vertex Coordinates

In \(y=a x^{2}\), the vertex form is \(y=a(x-h)^2+k\). Since \(h\) and \(k\) in this case are both 0, the coordinates of the vertex are (0,0).

Step 3: Equation of Axis of Symmetry

The axis of symmetry is a vertical line passing through the vertex (\(h, k\)). Here it is \(x = 0\).

Key Concepts

Vertex

The vertex of a quadratic function A vertex is a point on the graph of a quadratic function where it changes direction. In simpler terms, it's like the tip or the turning point of the parabola.

• When the parabola opens upwards, the vertex is the lowest point, known as the minimum.
• When the parabola opens downwards, the vertex is the highest point, known as the maximum.

For the quadratic function given \( y = -7x^{2} \), the vertex is found using the vertex form of a quadratic equation: \( y = a(x-h)^{2} + k \). Here, both \( h \) and \( k \) are zero, hence the vertex is at (0, 0). This means our parabola starts turning right at the origin.
To identify the vertex quickly, notice that in any function of the form \( y = ax^2 \), when there are no additional terms, the vertex is always at the origin.

Axis of Symmetry

The axis of symmetry in quadratic functions The axis of symmetry is a vertical line that runs through the vertex of the parabola. It divides the parabola into two mirror-image halves.

• For any parabola represented by the quadratic function \( y = ax^2 + bx + c \), the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \).
• In simpler functions like \( y = ax^2 \), which do not have \( bx \, or \,c \) terms, the axis of symmetry is simply \( x = 0 \).

In our case, we have \( y = -7x^2 \). This parabola is symmetric about the y-axis, thus its axis of symmetry is the vertical line \( x = 0 \).
This means, regardless of how wide or narrow the parabola, it will always have a line of symmetry exactly down the center where \( x = 0 \). Identifying the axis is essential as it helps in plotting the graph accurately and understanding its properties.

Direction of Parabola

The direction of a parabola's opening The direction of a parabola refers to whether it opens upwards or downwards. This is crucial in determining the nature of the function's graph.

• An upward-opening parabola has a positive \( a \) value in the quadratic function \( y = ax^2 \).
• A downward-opening parabola has a negative \( a \) value.

In the given function \( y = -7x^2 \), since the coefficient of \( x^2 \) is \(-7\), which is negative, the parabola opens downward.
This implies that as \( x \) increases or decreases, the value of \( y \) will continually decrease, creating a U-shape facing down. This is unlike functions with a positive \( a \), where \( y \) would increase symmetrically as \( x \) moves away from the vertex. Understanding the direction helps in predicting the behavior of the graph.