Problem 32 - Algebra for College Students

Question

Find the vertex for the graph of each quadratic function. $$y=x^{2}+8 x-3$$

Step-by-Step Solution

Answer

The vertex is (-4, -19).

Step 1 - Identify the coefficients

For the quadratic function in the form of: \[ y = ax^2 + bx + c \], identify the coefficients a, b, and c. In this case, \( a = 1 \), \( b = 8 \), and \( c = -3 \).

Step 2 - Use the vertex formula

The x-coordinate of the vertex for a quadratic function \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \]. Substituting the values of a and b: \[ x = -\frac{8}{2 \times 1} = -4 \].

Step 3 - Find the y-coordinate

To find the y-coordinate of the vertex, substitute the x-coordinate back into the original quadratic function: \( y = (-4)^2 + 8(-4) - 3 \) \( y = 16 - 32 - 3 \) \( y = -19 \).

Step 4 - State the vertex

Combine the x and y coordinates to give the vertex of the quadratic function. The vertex is (-4, -19).

Key Concepts

vertex formula

Let's start with understanding the vertex formula. The vertex of a quadratic function is the highest or lowest point on its graph, known as a parabola. This can be useful for finding the maximum or minimum value of the function. Given a quadratic function in the general form \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be found using the vertex formula: \[-\frac{b}{2a}\]. Here, \(a\) and \(b\) are coefficients of the quadratic and linear terms respectively. Once you have the x-coordinate, you can find the y-coordinate by substituting this x value back into the original equation. For instance, if \(y = x^2 + 8x - 3,\) the vertex formula gives us \(x = -\frac{8}{2} = -4\). Plugging \(x = -4\) back into the equation gives us the y-coordinate. Hence, the vertex is \((-4, -19)\).

quadratic function

A quadratic function is any function that can be written in the form \(y = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants, and \(a eq 0\). The graph of a quadratic function is a parabola, which can either open upwards (if \(a > 0\)) or downwards (if \(a < 0\)). The parabola's shape is determined by the value of \(a\), its position can be shifted based on values of \(b\) and \(c\). Key features of a quadratic function's graph include:

• Vertex, which is the highest or lowest point
• Axis of symmetry, a vertical line passing through the vertex
• Y-intercept, where the graph crosses the y-axis
• Finding these features help in graphing or understanding the quadratic function better.

coefficients in a quadratic equation

The coefficients \(a, b,\) and \(c\) in a quadratic equation \(y = ax^2 + bx + c\) have specific roles:

• Coefficient \(a\): Determines the parabola's direction and width. If \(|a| > 1,\) the parabola is narrower. If \(|a| < 1,\) it is wider. Positive \(a\) values make the parabola open upwards, while negative \(a\) values make it open downwards.
• Coefficient \(b\): Influences the position of the axis of symmetry and the vertex. It affects where the peak or trough of the parabola occurs horizontally.
• Coefficient \(c\): Represents the y-intercept, where the parabola crosses the y-axis. The value of \(c\) gives the function's starting point when \(x = 0\).

In the given exercise \(y = x^2 + 8x - 3\), \(a = 1\), \(b = 8\), and \(c = -3\). Identifying these coefficients helps in easily finding the vertex, axis of symmetry, and other characteristics of the quadratic function.

completing the square

Completing the square is another method to find the vertex of a quadratic function. It involves rewriting the quadratic function in vertex form \(y = a(x-h)^2 + k\), where \(h\) and \(k\) are the coordinates of the vertex. Here’s how you do it:

• Start with the standard form \(y = ax^2 + bx + c\).
• Factor out \(a\) from the first two terms: \(y = a(x^2 + \frac{b}{a}x) + c\).
• Add and subtract \(\left( \frac{b}{2a} \right)^2\) inside the parenthesis: \(y = a(x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2) + c\).
• Rewrite the perfect square trinomial: \(y = a\left( x + \frac{b}{2a} \right)^2 - \frac{b^2}{4a} + c\).
• Simplify to identify \(h = -\frac{b}{2a}\) and \(k = c - \frac{b^2}{4a}\).

This method can be seen as an alternate to applying the vertex formula directly. The answer will be the same, giving the vertex as \((-4, -19)\). This approach ensures a deeper understanding of how quadratic functions and their graphs work.