Problem 17 - Elementary Algebra

Question

Find the vertex and the line of symmetry. $$ y=-x 2+10 x-34 $$

Step-by-Step Solution

Answer

The vertex is \( (5, -9) \) and the line of symmetry is \( x = 5 \).

Step 1: Write the equation in standard form

The given quadratic equation is expressed as a standard quadratic equation which generally takes the form \( ax^2 + bx + c \). Here, rewrite the equation as \( y = -x^2 + 10x - 34 \), where \( a = -1 \), \( b = 10 \), and \( c = -34 \).

Step 2: Find the Line of Symmetry

The formula for the line of symmetry of a quadratic equation \( ax^2 + bx + c \) is \( x = -\frac{b}{2a} \). Substitute \( a = -1 \) and \( b = 10 \) into this formula: \[ x = -\frac{10}{2(-1)} = 5 \]. Thus, the line of symmetry is \( x = 5 \).

Step 3: Calculate the Vertex x-coordinate

Since we already found the line of symmetry as \( x = 5 \), the x-coordinate of the vertex is \( x = 5 \).

Step 4: Calculate the Vertex y-coordinate

To find the y-coordinate of the vertex, substitute \( x = 5 \) into the original equation \( y = -x^2 + 10x - 34 \): \( y = -(5)^2 + 10(5) - 34 = -25 + 50 - 34 = -9 \). So, the y-coordinate of the vertex is \( -9 \).

Step 5: Express the Vertex

The vertex of the parabola, given by the x and y coordinates found above, is \( (5, -9) \).

Key Concepts

vertex of a parabola

The vertex of a parabola is a crucial point that represents either the highest or lowest point on the graph, depending on its orientation. In mathematical terms, it is a point \( (h, k) \) where the parabola changes direction. This transformation point is important because it helps in understanding the parabola's shape and position in a coordinate plane.
The vertex can be found using the formula for the line of symmetry, since the x-coordinate of the vertex is always located on this line. In the equation for the parabola \( y = ax^2 + bx + c \), once the x-coordinate \( x = -\frac{b}{2a} \) is determined, substitute this value back into the original equation to find the y-coordinate.

• For the equation \( y = -x^2 + 10x - 34 \), we already found the vertex's x-coordinate to be 5.
• Substitute this back to get: \( y = -(5)^2 + 10(5) - 34 = -25 + 50 - 34 = -9 \).
• Hence, the vertex is \( (5, -9) \).

Understanding the vertex helps in graphing the parabola easily and tackling many real-world problems that can be modeled by quadratic equations.

line of symmetry

The line of symmetry in a parabola is an imaginary vertical line that divides the parabola into two mirror-image halves. It passes through the vertex of the parabola and allows us to identify the parabola's axis of symmetry.
This line of symmetry is calculated using the formula \( x = -\frac{b}{2a} \) for quadratic equations in the form \( ax^2 + bx + c \). This formula helps in pinpointing where the vertex lies on the x-axis.

• For the equation \( y = -x^2 + 10x - 34 \), substitute \( a = -1 \) and \( b = 10 \) into the formula to get:
• \( x = -\frac{10}{2(-1)} = 5 \).
• This indicates that the line of symmetry is \( x = 5 \).

Identifying the line of symmetry simplifies the graphing process and is vital in solving many quadratic problems efficiently.

standard form of a quadratic equation

The standard form of a quadratic equation is expressed as \( ax^2 + bx + c = 0 \), where "a", "b", and "c" are constants, and \( a \) is not zero. This form is fundamental as it helps automatically outline the basic structure of a quadratic equation.
To work effectively with quadratic equations, it's essential to recognize and rearrange equations into this format, which aids in the subsequent steps for solving or graphing.

• In our example, the equation is initially given as \( y = -x^2 + 10x - 34 \), which is already in the standard form since:
• \( a = -1 \), \( b = 10 \), and \( c = -34 \).

Being comfortable with manipulating equations into standard form is key to accurately applying formulas like the quadratic formula, determining the vertex, and finding the line of symmetry.