Problem 75 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the graph of the function and identify the vertex. $$f(x)=x^{2}-7 x-8$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=x^{2}-7x-8\) is a parabola opening upwards with its vertex at the point \((3.5, -14.75)\).

Step 1: Identifying The Format of The Equation

Here, we have a quadratic function in the standard form, \(f(x)=x^{2}-7x-8\). The standard form of a quadratic function is \(ax^2 + bx + c\), where \(a, b, c\) are constants, and \(a ≠ 0\). In our function, \(a=1\), \(b=-7\), and \(c=-8\).

Step 2: Computing The Vertex

The vertex \((h, k)\) of a parabola \(y = ax^2 + bx + c\) can be found using the formula \(h = -b/2a\) and \(k = f(h)\). Therefore, the x-coordinate of the vertex is \(h = -(-7)/(2*1) = 3.5\). Substituting \(3.5\) into the function gives \(k = f(3.5) = (3.5)^{2}-7*(3.5)-8 = -14.75\). Therefore, the vertex of the quadratic function is \((3.5, -14.75)\).

Step 3: Describing The Graph

The graph of the function is a parabola opening upwards (since \(a =1 > 0\)) with a vertex at the point \((3.5,-14.75)\). It is shifted 3.5 units to the right and 14.75 units down from the origin. And because it opens upwards, the vertex \((3.5, -14.75)\) is the minimum point on the graph.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is one of the most critical points on its graph. It is either the lowest or highest point depending on whether the parabola opens upwards or downwards. For the quadratic function given by the standard form equation \(y = ax^2 + bx + c\), the vertex can be found using a specific formula. The x-coordinate of the vertex is calculated using \(h = -\frac{b}{2a}\), and the y-coordinate or \(k\), is found by substituting this x-value into the function.
For our example, \(f(x) = x^2 - 7x - 8\), the vertex is at \((3.5, -14.75)\). This means the vertex is 3.5 units along the x-axis and 14.75 units below the x-axis, making it the minimum point of the parabola since our parabola opens upwards.

Graphing Parabolas

Graphing a parabola requires an understanding of its shape and direction. A parabola is a symmetric curve with a peak or dip at the vertex. The direction it opens is determined by the sign of the coefficient \(a\). If \(a > 0\), the parabola opens upwards, like a smile. If \(a < 0\), it opens downwards, like a frown.
To graph one, start by plotting the vertex, then use additional points on either side of the vertex to complete the curve.
The axis of symmetry, given by \(x = h\), splits the parabola into two mirror-image halves. In this case, the axis is \(x = 3.5\). Plot points equidistant from this axis to ensure an accurate graph.

• The vertex is the focal point of the graph.
• Additional points help define the width and shape.
• Symmetry around the vertex ensures balance.

Quadratic Equation Standard Form

The standard form of a quadratic equation is essential for easily identifying the parabola's properties. This equation is written as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\).
This form is especially useful because it allows direct computation of the vertex, the axis of symmetry, and helps determine the parabola's direction.

• The coefficient \(a\) affects the parabola's width and direction.
• The coefficient \(b\) influences the parabola's tilt.
• \(c\) gives the y-intercept, a point where the parabola crosses the y-axis.

By comparing a given quadratic equation to the standard form, you can quickly identify these values and understand the graph's general behavior. This quick comparison and computation are invaluable tools for graphing and interpreting quadratic functions.