Problem 4 - Algebra 2

Question

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=x^{2}+8 x+3 $$

Step-by-Step Solution

Answer

The vertex is at \((-4, -13)\), the y-intercept is 3, and the axis of symmetry is \(x = -4\).

Step 1: Find the y-intercept

The y-intercept of a quadratic function is found by substituting \( x = 0 \) in the equation. For the function \( f(x) = x^2 + 8x + 3 \), we calculate \( f(0) = 0^2 + 8 imes 0 + 3 = 3 \). Thus, the y-intercept is 3.

Step 2: Find the equation of the axis of symmetry

The axis of symmetry for a quadratic function in the form \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). For the function \( f(x) = x^2 + 8x + 3 \), \( a = 1 \) and \( b = 8 \). Thus, the axis of symmetry is \( x = -\frac{8}{2 \times 1} = -4 \).

Step 3: Find the x-coordinate of the vertex

The x-coordinate of the vertex is the same as the axis of symmetry: \( x = -4 \). Thus, the vertex's x-coordinate is \( -4 \).

Step 4: Find the y-coordinate of the vertex

To find the y-coordinate, substitute the x-coordinate of the vertex into the function: \( f(-4) = (-4)^2 + 8(-4) + 3 = 16 - 32 + 3 = -13 \). Therefore, the vertex is \((-4, -13)\).

Step 5: Make a table of values including the vertex

Choose values around the vertex to include in a table. For example: \((-6, 7)\), \((-5, -7)\), \((-4, -13)\), \((-3, -7)\), and \((-2, 3)\). These points will help plot the function.

Step 6: Use information to graph the function

Plot the points from the table, especially the vertex, and draw a parabola opening upward based on the pattern of these points.

Key Concepts

Axis of Symmetry

The axis of symmetry is a line that splits the parabola into two mirror-image halves. For any quadratic function in the form of \( ax^2 + bx + c \), it can be easily determined by using the formula \( x = -\frac{b}{2a} \). This formula provides the x-coordinate of the vertex, which is the point where the parabola is closest to the axis of symmetry.
In the function \( f(x) = x^2 + 8x + 3 \), we find \( a = 1 \) and \( b = 8 \). Substituting these values into the formula gives us the axis of symmetry as \( x = -\frac{8}{2 \times 1} = -4 \). This vertical line at \( x = -4 \) is crucial, as it indicates where to fold the graph to line up the two sides perfectly.

Vertex of a Parabola

The vertex of a parabola is a key feature because it is the turning point or the "tip" of the parabola. For a quadratic function like \( f(x) = x^2 + 8x + 3 \), the vertex can be found using the axis of symmetry. Since we know the x-coordinate from the axis of symmetry to be \( x = -4 \), we can find the y-coordinate by substituting \( -4 \) into the function:

• Calculate \( f(-4) = (-4)^2 + 8(-4) + 3 = 16 - 32 + 3 = -13 \).

Thus, the vertex is at the point \((-4, -13)\).
The vertex not only informs us about the highest or lowest point of the graph depending on whether the parabola opens upwards or downwards, but also helps in graphing the parabola accurately.

Y-Intercept

The y-intercept is the point where the graph intersects the y-axis. It represents the value of the function when \( x = 0 \). To find it, you simply substitute \( x = 0 \) into the equation,

• Given the function \( f(x) = x^2 + 8x + 3 \), substitute to find \( f(0) = 0^2 + 8 \times 0 + 3 = 3 \).

The y-intercept therefore is \( 3 \), meaning the parabola crosses the y-axis at the point \((0, 3)\).
Understanding the y-intercept is essential for sketching the graph correctly because it provides a starting point on the y-axis.

Graphing Quadratics

Graphing quadratic functions involves plotting a few key points including the vertex and y-intercept. This provides a skeletal framework to sketch the shape of the parabola. Here are a few steps to graph the quadratic function effectively:

• Identify the vertex, which is \((-4, -13)\) for this function.
• Find the y-intercept, found to be \((0, 3)\).
• Take some points around the vertex from the table, such as \((-6, 7)\), \((-5, -7)\), and \((-2, 3)\)

Begin plotting these points on a graph paper.
Connect the dots in a smooth curve, ensuring the parabola opens upwards because the coefficient of \( x^2 \) is positive. The curve should look symmetric on either side of the vertex, and the axis of symmetry should guide this alignment.
By following these steps, you can graph any quadratic function with confidence, understanding how each part relates to the others in forming the overall shape of the graph.