Problem 14 - Intermediate Algebra

Question

Graph each of the following linear and quadratic functions. $$f(x)=-x^{2}-8 x-15$$

Step-by-Step Solution

Answer

The vertex is at (-4, 1), y-intercept at (0, -15), and x-intercepts at (-3, 0) and (-5, 0). The parabola opens downward.

Step 1: Understanding the Function

The given function is a quadratic function of the form \(f(x) = ax^2 + bx + c\), where \(a = -1\), \(b = -8\), and \(c = -15\). This is a downward opening parabola because \(a < 0\).

Step 2: Find the Vertex

For a quadratic function \(f(x) = ax^2 + bx + c\), the vertex can be found using the formula \(x = -\frac{b}{2a}\). Substitute \(b = -8\) and \(a = -1\) into the formula: \(x = -\frac{-8}{2(-1)} = -4\). Substitute \(x = -4\) back into the function to find \(f(-4) = -(-4)^2 - 8(-4) - 15 = -16 + 32 - 15 = 1\). Thus, the vertex is \((-4, 1)\).

Step 3: Find the Y-intercept

The y-intercept of the function occurs where \(x=0\). Substitute \(x=0\) into the function: \(f(0) = -(0)^2 - 8(0) - 15 = -15\). Thus, the y-intercept is \((0, -15)\).

Step 4: Find the X-intercepts

To find the x-intercepts, set \(f(x)\) to zero and solve for \(x\):\(-x^2 - 8x - 15 = 0\). Multiply the entire equation by \(-1\) to simplify: \(x^2 + 8x + 15 = 0\). Factor the quadratic: \((x + 3)(x + 5) = 0\). Thus, \(x + 3 = 0\) or \(x + 5 = 0\), giving solutions \(x = -3\) and \(x = -5\). These are the x-intercepts: \((-3, 0)\) and \((-5, 0)\).

Step 5: Plot and Analyze the Graph

Now plot the points found: the vertex \((-4, 1)\), the y-intercept \((0, -15)\), and the x-intercepts \((-3, 0)\) and \((-5, 0)\). Draw a parabola that opens downward, passing through all these points. The vertex is the highest point because the parabola opens downward due to the negative \(a\) value.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open either upwards or downwards. In quadratic functions, the parabola's orientation is determined by the coefficient of the squared term, denoted as ​\(a\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards. This shape is fundamental to realize in graphing because it dictates the general direction of the parabola's arms. When analyzing a parabola like \(f(x) = -x^2 - 8x - 15\), we recognize immediately that it opens downwards because \(a = -1\), which is less than zero.- **Shape:** U-shaped curve- **Direction:** Depends on the sign of \(a\)Visualizing the parabola on the graph helps identify crucial characteristics such as symmetry, and it can visually guide you when locating its vertex and intercepts.

Vertex of a Quadratic Function

The vertex of a quadratic function is a key point located at the peak or the lowest point of the parabola, depending on whether it opens up or down. This is the turning point where the parabola changes direction. To find the vertex in the standard quadratic form, \(ax^2 + bx + c\), use the vertex formula \(x = -\frac{b}{2a}\). For example, in the function \(f(x) = -x^2 - 8x - 15\), substitute:

• \(b = -8\) and \(a = -1\) into the formula to get \(x = -\frac{-8}{2(-1)} = -4\).
• Find the y-coordinate by substituting \(x = -4\) back into the function: \(f(-4) = 1\).
• Thus, the vertex is \((-4, 1)\).

This vertex represents the highest point on this parabola because it opens downward. Always remember to calculate both x and y coordinates for the vertex to accurately graph the parabola.

X-intercepts and Y-intercepts

Understanding x-intercepts and y-intercepts of a quadratic function is essential for sketching the graph of a parabola. These intercepts mark where the parabola crosses the x-axis and y-axis respectively.**Y-intercept:**- The y-intercept occurs where \(x = 0\).- Substitute into the function to find the point: For \(f(x) = -x^2 - 8x - 15\), setting \(x\) to zero gives \(f(0) = -15\).- Thus, the y-intercept is \((0, -15)\).**X-intercepts:**- These are points where the function equals zero, \(f(x) = 0\).- Solve \(-x^2 - 8x - 15 = 0\) by first multiplying the equation by -1 to simplify: \(x^2 + 8x + 15 = 0\).- Factor the quadratic: \((x + 3)(x + 5) = 0\). - Therefore, \(x = -3\) and \(x = -5\), the x-intercepts are \((-3, 0)\) and \((-5, 0)\).These intercepts provide crucial points for plotting the parabola, helping outline its path on the graph and reinforcing the understanding of how the quadratic function behaves.