Problem 20 - 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. $$ f(x)=x^{2}-4 x-5 $$

Step-by-Step Solution

Answer

Vertex is (2, -9), y-intercept is (0, -5), axis of symmetry is x = 2.

Step 1: Find the y-intercept

The y-intercept of a quadratic function is found by evaluating the function at \( x = 0 \). For the function \( f(x) = x^2 - 4x - 5 \), substitute \( x = 0 \):\[f(0) = 0^2 - 4(0) - 5 = -5\]Thus, the y-intercept is \( (0, -5) \).

Step 2: Find the Axis of Symmetry Equation

The equation for the axis of symmetry of a quadratic function \( f(x) = ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). For \( f(x) = x^2 - 4x - 5 \), we have \( a = 1 \) and \( b = -4 \):\[x = -\frac{-4}{2(1)} = 2\]So, the axis of symmetry is \( x = 2 \).

Step 3: Find the x-coordinate of the Vertex

The x-coordinate of the vertex is the same as the equation of the axis of symmetry, \( x = 2 \).

Step 4: Find the y-coordinate of the Vertex

To find the y-coordinate of the vertex, substitute \( x = 2 \) into the function \( f(x) = x^2 - 4x - 5 \):\[f(2) = 2^2 - 4(2) - 5 = 4 - 8 - 5 = -9\]Thus, the vertex is \( (2, -9) \).

Step 5: Create a Table of Values

Create a table of values for \( x \) including the vertex \( (2, -9) \) and other points around it. Let's choose \( x = 0, 1, 2, 3, 4 \):\[\begin{array}{c|c}x & f(x) \\hline0 & -5 \1 & 1^2 - 4(1) - 5 = -8 \2 & -9 \3 & 3^2 - 4(3) - 5 = -8 \4 & 16 - 16 - 5 = -5\end{array}\]So, we have the points: \((0, -5), (1, -8), (2, -9), (3, -8), (4, -5)\).

Step 6: Plot the Graph

Using the information gathered, you can plot the graph of \( f(x) = x^2 - 4x - 5 \). Plot the vertex \( (2, -9) \), the y-intercept \( (0, -5) \), and the additional points \((1, -8)\), \((3, -8)\), and \((4, -5)\). Draw a smooth curve through these points to complete the parabolic graph.

Key Concepts

Vertex

The vertex of a quadratic function is a crucial component because it represents either the highest or lowest point on the graph of the function. For a quadratic function in the form \[f(x) = ax^2 + bx + c\]to find the vertex:

• The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \). This formula locates the line of symmetry of the parabola.
• In the given function \( f(x) = x^2 - 4x - 5 \), applying the formula, we find the x-coordinate of the vertex is \( x = 2 \).
• Substitute \( x = 2 \) back into the function to find the y-coordinate; thus, \( f(2) = 2^2 - 4(2) - 5 = -9 \).

So, the vertex is \((2, -9)\). This position helps in understanding the graph's turning point, making it a point of maximum or minimum depending on the parabola's direction.

Axis of Symmetry

The axis of symmetry for a quadratic function is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex, offering a simplistic view of the parabola's balance.

• The formula to find the axis of symmetry for any quadratic function \( f(x) = ax^2 + bx + c \) is \( x = -\frac{b}{2a} \).
• In our function, \( f(x) = x^2 - 4x - 5 \), plug in the values: \( x = -\frac{-4}{2 \times 1} = 2 \).

This information indicates that the axis of symmetry is the line \( x = 2 \). Knowing this allows us to predict that points equidistant from this line will yield equal values of \( f(x) \). It aids in creating an even plot across the graph.

Y-intercept

The y-intercept of a quadratic function is the point where the graph intersects the y-axis. This is found when the input or \( x \)-value is 0.

• Substitute \( x = 0 \) into the equation \( f(x) = x^2 - 4x - 5 \). Calculating gives \( f(0) = 0^2 - 4(0) - 5 = -5 \).
• Thus, the y-intercept is \((0, -5)\).

This point is significant as it serves as a starting point for plotting the graph along with the vertex. The y-intercept helps in understanding how the graph behaves near the y-axis and provides a benchmark for sketching it precisely. Notably, every quadratic graph will have exactly one y-intercept.