Problem 9 - Thinking Mathematically

Question

In Exercises 9-14, a. Determine if the parabola whose equation is given opens upward or downward. b. Find the vertex. c. Find the \(x\)-intercepts. d. Find the y-intercept. e. Use (a)-(d) to graph the quadratic function. \(y=x^{2}+8 x+7\)

Step-by-Step Solution

Answer

The parabola opens upward, the vertex is at (-4, -9), the \(x\)-intercepts are at \(x = -1\) and \(x = -7\), the \(y\)-intercept is at \(y = 7\) and the graph of the quadratic equation is an upward-opening parabola with these points.

Step 1: Determine the direction

The equation given is \(y = x^2 + 8x + 7\). The coefficient of \(x^2\) is 1, which is positive. Therefore, the parabola opens upward.

Step 2: Find the vertex

Use the formula \(-b/2a\) to find the \(x\)-value of the vertex. Here, \(a = 1\) and \(b = 8\), so \(-b/2a = -8/2*1 = -4\). Substitute \(-4\) into the equation for \(x\) to find the \(y\)-value. \(y = (-4)^2 + 8*(-4) + 7 = -9\). Therefore, the vertex is at (-4, -9).

Step 3: Find the \(x\)-intercepts

To find the \(x\)-intercepts, set \(y = 0\) and solve for \(x\). \(0 = x^2 + 8x + 7\). Factoring the equation gives \(0 = (x + 1)(x + 7)\). Thus, \(x = -1\) and \(x = -7\) are the \(x\)-intercepts.

Step 4: Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation: \(y = 0^2 + 8*0 + 7 = 7\). Thus, the \(y\)-intercept is at \(y = 7\).

Step 5: Graph the quadratic function

Plot the vertex, \(x\)-intercepts, and \(y\)-intercept on the graph. Sketch the parabola, making sure it opens upwards.

Key Concepts

Parabolas

Understanding the concept of parabolas is essential when studying quadratic functions. A parabola is a U-shaped graph that can either open upwards or downwards, and it represents the visual aspect of a quadratic equation. In our study of the quadratic function y = x^2 + 8x + 7, it's notable that the coefficient of the x^2 term determines the direction of the parabola. Since the coefficient here is positive, our parabola gracefully opens upward like a welcoming arm.

When graphing, the parabola's curvature will change depending on the magnitude of this coefficient; the larger it is, the more 'narrow' the arms of the parabola become. Conversely, a smaller positive coefficient results in a wider shape. The presence of the x and constant terms can shift and stretch the graph, but the fundamental U-shape remains the same.

Vertex of a Parabola

The vertex of a parabola holds significant importance because it represents the peak or the lowest point on the graph for upward and downward opening parabolas, respectively. For the function y = x^2 + 8x + 7, we identify the vertex using a simple formula, \( -b/2a \). Here, 'a' and 'b' are coefficients from the standard form of a quadratic equation ax^2 + bx + c.

For our example, a = 1 and b = 8, which, when plugged into the formula, gives us the x coordinate of the vertex at \( x = -4 \). To find the corresponding y coordinate, we substitute \( x = -4 \) back into the equation to receive \( y = -9 \), completing our vertex point at \( (-4, -9) \). This point is crucial when sketching the parabola as it defines the symmetry and the turnaround point of the graph.

X-Intercepts

The x-intercepts of a parabola, often referred to as the roots or solutions of the quadratic equation, are the points where the graph intersects the x-axis. These points are found by setting the output y to zero and solving the resulting equation. In our quadratic function y = x^2 + 8x + 7, we achieve this by determining the values of x that make the equation equal to zero.

Through factorization, we can split the equation into \( (x + 1)(x + 7) = 0 \), which provides us with two potential solutions: \( x = -1 \) and \( x = -7 \). These values are the x-coordinates for the points where our parabola meets the x-axis, helping to anchor the graph further.

Y-Intercept

The y-intercept of a parabola is the point where the graph crosses the y-axis, and it's found by evaluating the function when x is zero. For the function in question, y = x^2 + 8x + 7, setting x to zero simplifies the equation to \( y = 0^2 + 8*0 + 7 \), resulting in \( y = 7 \).

This sole y-intercept, (0, 7), provides a starting point for the graph on the y-axis. Together with the vertex and x-intercepts, the y-intercept assists in shaping the overall structure of the parabola, contributing to a complete graphical representation of the quadratic function.