Problem 32
Question
Describe the graph of the quadratic function. Identify the vertex and \(x\) -intercept(s). Use a graphing utility to verify your results. \(f(x)=x^{2}+10 x+14\)
Step-by-Step Solution
Verified Answer
The vertex of the function is at \((-5, -11)\) and the x-intercepts are at \((-8, 0)\) and \((-2, 0)\).
1Step 1: Finding the Vertex
Use the formula \(-b / 2a\) to find the x-coordinate of the vertex, \(h\). Here, \(b = 10\) and \(a = 1\). Therefore, \(h = -10 / (2 * 1) = -5\). Plug \(h\) into the equation to find the y-coordinate of the vertex, \(k = f(-5)\). This means \(k = (-5)^{2} +10*(-5) + 14 = -11\). So, the vertex is \((-5, -11)\).
2Step 2: Finding the x-intercepts
Use the quadratic formula \(x = [-b ± sqrt(b^2-4ac)]/2a\). Here \(b = 10\), \(a = 1\) and \(c = 14\). So, \(x = [-10 ± sqrt((10)^2-4*1*14)]/(2*1) = [-10 ± sqrt(36)]/2 = [-10 ± 6]/2\). This gives two solutions: \(x1 = -8\) and \(x2 = -2\). These are the x-intercepts.
3Step 3: Verifying the Results Using a Graphing Utility
Plot the function \(f(x)=x^{2}+10 x+14\), the vertex \((-5, -11)\) and the x-intercepts \((-8, 0)\) and \((-2, 0)\) using a graphing utility. The vertex is the lowest point of the graph (as it's a upward opening parabola) and the graph intersects x-axis at the x-intercepts.
Key Concepts
Vertex of a ParabolaQuadratic FormulaX-interceptsGraphing Quadratic Functions
Vertex of a Parabola
The vertex of a parabola represents the highest or lowest point on the graph, depending on whether it opens upward or downward. For the quadratic function in standard form, given by \(f(x)=ax^{2}+bx+c\), you can find the vertex using the formula \(h=-\frac{b}{2a}\) for the x-coordinate, and then substitute \(h\) back into the function to find the y-coordinate, which can be represented as \(k=f(h)\).
In our exercise, we have \(a=1\), \(b=10\), and \(c=14\), which gives us a vertex at the point \((-5,-11)\). This means the parabola reaches its minimum value at \(x=-5\), and the value of the function at this point is \(-11\).
It's critical to understand that the direction a parabola opens (upward or downward) is determined by the leading coefficient \(a\). Since \(a=1\) in this case, which is positive, our parabola opens upward, making the vertex a minimum point.
In our exercise, we have \(a=1\), \(b=10\), and \(c=14\), which gives us a vertex at the point \((-5,-11)\). This means the parabola reaches its minimum value at \(x=-5\), and the value of the function at this point is \(-11\).
It's critical to understand that the direction a parabola opens (upward or downward) is determined by the leading coefficient \(a\). Since \(a=1\) in this case, which is positive, our parabola opens upward, making the vertex a minimum point.
Quadratic Formula
Solving quadratic equations can be achieved through the quadratic formula, which is derived from the process of completing the square on the general quadratic equation \(ax^{2} + bx + c = 0\). The quadratic formula is given by \(x = \frac{{-b \pm \sqrt{b^{2} - 4ac}}}{2a}\).
Here, \(b\) and \(a\) are the coefficients from the quadratic function, and \(c\) is the constant term. This formula calculates the roots of the quadratic function — in other words, the x-intercepts. The term under the square root, \(b^2 - 4ac\), is known as the discriminant, and it is essential in determining the number and type of roots the equation will have. A positive discriminant means two real roots, zero means one real root (also known as a repeated or double root), and a negative discriminant means there are no real roots (the roots are complex numbers).
Applied to our exercise, using the values \(b=10\), \(a=1\), and \(c=14\), we find the x-intercepts at \(-8\) and \(-2\), showing where the graph crosses the x-axis.
Here, \(b\) and \(a\) are the coefficients from the quadratic function, and \(c\) is the constant term. This formula calculates the roots of the quadratic function — in other words, the x-intercepts. The term under the square root, \(b^2 - 4ac\), is known as the discriminant, and it is essential in determining the number and type of roots the equation will have. A positive discriminant means two real roots, zero means one real root (also known as a repeated or double root), and a negative discriminant means there are no real roots (the roots are complex numbers).
Applied to our exercise, using the values \(b=10\), \(a=1\), and \(c=14\), we find the x-intercepts at \(-8\) and \(-2\), showing where the graph crosses the x-axis.
X-intercepts
X-intercepts are the points where the graph of a function crosses the x-axis. For a quadratic function, these points correspond to the roots, or solutions, of the equation \(ax^{2}+bx+c=0\). You can find x-intercepts by setting the function equal to zero and solving for \(x\).
While factoring and completing the square are valid methods for finding the x-intercepts, the quadratic formula is often the most straightforward approach when the equation cannot be easily factored. In our example, the x-intercepts were found to be at \(x = -8\) and \(x = -2\), indicating two distinct points where the parabola intersects the x-axis. These intercepts are crucial for graphing the function and understanding its behavior.
While factoring and completing the square are valid methods for finding the x-intercepts, the quadratic formula is often the most straightforward approach when the equation cannot be easily factored. In our example, the x-intercepts were found to be at \(x = -8\) and \(x = -2\), indicating two distinct points where the parabola intersects the x-axis. These intercepts are crucial for graphing the function and understanding its behavior.
Graphing Quadratic Functions
To graph a quadratic function, one must understand the general shape of a parabola and the function's defining features, such as the vertex and x-intercepts. The function is usually given in the form \(f(x) = ax^{2} + bx + c\), where \(a\) determines the direction of the opening, and the vertex provides a starting point for the graph.
Graphing starts by plotting the vertex, then the x-intercepts, and then plotting additional points as necessary by choosing suitable x-values and calculating their corresponding y-values. The axis of symmetry, a vertical line that passes through the vertex, can also guide the graphing process. By reflecting points across the axis of symmetry, one ensures the parabola is accurate and symmetrical.
Using technology, such as graphing calculators or software, can assist in verifying the results. In our problem, a graphing utility would show the vertex at \((-5, -11)\) and the x-intercepts at \((-8, 0)\) and \((-2, 0)\), confirming the accuracy of our calculations.
Graphing starts by plotting the vertex, then the x-intercepts, and then plotting additional points as necessary by choosing suitable x-values and calculating their corresponding y-values. The axis of symmetry, a vertical line that passes through the vertex, can also guide the graphing process. By reflecting points across the axis of symmetry, one ensures the parabola is accurate and symmetrical.
Using technology, such as graphing calculators or software, can assist in verifying the results. In our problem, a graphing utility would show the vertex at \((-5, -11)\) and the x-intercepts at \((-8, 0)\) and \((-2, 0)\), confirming the accuracy of our calculations.
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