Question

passes through \((-7,-15)\) and has vertex \((-5,9)\)

Short answer

The equation of the parabola that passes through the point (-7,-15) and has vertex (-5,9) is \(y=-1.5(x+5)^2 + 9\).

Step-by-step solution

Identify the Equation Format

The Equation of a parabola in vertex form is given by \(y=a(x-h)^2 + k\) where \((h,k)\) are the coordinates of the vertex.

Substitute the Vertex into the Equation

Substitute the given vertex (-5,9) for (h,k) in the equation, Now, equation becomes \(y=a(x+5)^2 + 9\)

Solving for 'a' using the given point

Given, parabola passes through the point (-7,-15). Substitute these values for 'x' and 'y' to solve for 'a'. \n-15 = a(-7+5)^2 + 9 \n-15 = a(4)^2 + 9 \n-15=16a+9. Rearranging this and then dividing by 16, we find 'a' equals -1.5.

Write Down Final Equation

Replace 'a' in the equation with -1.5 to get the final equation of the parabola. \nSo, \(y=-1.5(x+5)^2 + 9\)

Key concepts

Algebra 2

In \textbf{Algebra 2}, students delve deeper into arithmetic operations and become adept at manipulating complex equations and functions. A key aspect is understanding the properties of various functions, including quadratics. Quadratics, or second-degree polynomials, take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \) and \( c \) are constants, and \( a \eq 0 \).

By mastering methods such as factoring, completing the square, or using the quadratic formula, one can solve these equations efficiently. The parabola, a u-shaped curve represented by a quadratic function, is central to Algebra 2. Its properties—such as the axis of symmetry, the vertex, and the direction of opening—are fundamental in understanding the graph's behavior and in solving related problems.

Vertex of a Parabola

The \textbf{vertex of a parabola} is the highest or lowest point on its graph, which occurs at the axis of symmetry. In the vertex form of a quadratic equation \( y = a(x-h)^2 + k \), the vertex is the point \( (h, k) \). Here, \( h \) and \( k \) give the location of the vertex on the \( x \) and \( y \) axes, respectively.

In the provided example, the vertex is given as \( (-5, 9) \), indicating that the parabola opens either upwards or downwards depending on the value of \( a \) and crosses the \( y \) axis at \( (0, k) \). Vertex form makes it simple to translate the parabola vertically or horizontally and to reflect it across the \( x \) or \( y \) axes, by adjusting \( h \) and \( k \) respectively.

Solving Quadratic Equations

Working with quadratic equations often requires finding their roots or solutions, which are the \( x \) values where the parabola crosses the \( x \) axis. There are different methods to \textbf{solve quadratic equations}, including graphing, using the quadratic formula, factoring, or completing the square.

When the quadratic equation is presented in vertex form, as in \( y = a(x-h)^2 + k \), we can plug in known coordinates to solve for \( a \) and thus find the specific equation of the parabola. Using the point \( (-7, -15) \) in the example helps us determine that the coefficient \( a \) is \( -1.5 \), thus giving us the precise stretch and direction of the parabola. When the value of \( a \) is negative, as in our example, it indicates that the parabola opens downward.