Problem 31 - 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. $$ y=-10 x^{2}+12 x $$

Step-by-Step Solution

Answer

a. The function's graph opens downwards. b. The vertex coordinates are (0.6, 2.4). c. The equation of the axis of symmetry is \(x=0.6\).

Step 1: Direction of Opening

The coefficient of the \(x^2\) term in the quadratic function - which is -10 - determines the direction in which the graph of the function opens. If it's negative, the function's graph opens downwards. Therefore, the graph of the function \(y=-10x^2 + 12x\) opens down.

Step 2: Find the Vertex

The coordinates of the vertex can be found using the formula for the x-coordinate, \(x = -b/2a\), where \(a\) and \(b\) are the coefficients of the \(x^2\) and \(x\) terms of the quadratic function, respectively. In this function, \(a = -10\) and \(b = 12\). Thus, \[x = -b/(2a) = -12/(2*-10) = 0.6\]. The y-coordinate is the value of the function at the x-coordinate of the vertex, so \(y = -10*(0.6)^2 + 12*0.6 = 2.4\). Therefore, the vertex coordinates are (0.6, 2.4).

Step 3: Equation of Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line passing through the vertex, and it can be written as \(x = h\), where \(h\) is the x-coordinate of the vertex. In this case the vertex coordinates are (0.6, 2.4), so the axis of symmetry is \(x = 0.6\).

Key Concepts

Axis of Symmetry

The axis of symmetry in a quadratic function is a crucial concept that helps in understanding the reflective symmetry of a parabola. It is a vertical line that goes through the vertex of the parabola, making it a mirror-like line where the left and right sides of the parabola are mirror images of each other.

To find the equation of the axis of symmetry, we use the formula \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the quadratic function in the standard form \( y = ax^2 + bx + c \).

• For the given function \( y=-10x^2+12x \), \( a = -10 \) and \( b = 12 \).
• Plugging these values in gives us \( x = -\frac{12}{2 \times -10} = 0.6 \).
• Therefore, the axis of symmetry for the function is the line \( x = 0.6 \).

Understanding and identifying the axis of symmetry is beneficial for graphing the function and finding the vertex, as the vertex lies on this axis.

Vertex of a Parabola

The vertex of a parabola is a major feature of the graph of a quadratic function. It represents the highest or lowest point on the graph, depending on the direction the parabola opens. For a function such as \( y = ax^2 + bx + c \), the vertex is at the point \( (h, k) \) where \( h \) is derived from the axis of symmetry equation \( x = -\frac{b}{2a} \) and \( k \) is the value of \( y \) when \( x = h \).

In the function \( y = -10x^2 + 12x \) given in the exercise:

• We already calculated \( h = 0.6 \) when finding the axis of symmetry.
• To find \( k \), we substitute \( h \) into the function to get \( k = -10(0.6)^2 + 12 \times 0.6 = 2.4 \).
• Thus, the vertex of the parabola is at (0.6, 2.4).

Identifying the vertex helps in sketching the parabola and understanding its properties like the maximum or minimum value of the function.

Direction of Opening

The direction of opening of a parabola tells us whether the arms of the parabola point upwards or downwards—or, simply put, whether the parabola opens up or down. This is directly determined by the sign of the coefficient \( a \) in the quadratic function's standard form \( y = ax^2 + bx + c \).

• If \( a > 0 \), the parabola opens upwards, resembling a 'U' shape, which indicates a minimum vertex.
• If \( a < 0 \), the parabola opens downwards, resembling an upside-down 'U' or an 'n' shape, indicating a maximum vertex.

For the exercise function \( y = -10x^2 + 12x \), \( a = -10 \), which is less than zero. Hence, the graph of this quadratic function opens downwards, and the vertex at (0.6, 2.4) represents the highest point on the graph. Understanding the direction of opening is vital for anticipating the behavior of the function, particularly in real-world applications where the parabola might represent trajectories, profit curves, or other phenomena.