Problem 10 - Thinking Mathematically

Question

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. \(y=x^{2}+10 x+9\)

Step-by-Step Solution

Answer

The parabola opens upward. Vertex is (-5,-16). X-intercepts are -1 and -9, and the y-intercept is at y=9.

Step 1: Determine direction

Since \(a=1\) which is greater than 0, the parabola opens upwards.

Step 2: Find the vertex

To find the vertex, calculate \(h=-\frac{b}{2a}=-\frac{10}{2}= -5\). Then find the y-coordinate of the vertex by substituting \(h=-5\) into the equation to get \(k = (-5)^{2}+10*(-5)+9 = -16\). Therefore, the vertex is (-5,-16).

Step 3: Find the x-intercepts

Set \(y=0\) and solve the equation for \(x\). \(0 = x^{2}+10x+9\). Factor the equation to get \(0=(x+1)(x+9)\). Set each factor equal to zero and solve for \(x\), \(x+1=0\) yields \(x=-1\) and \(x+9=0\) yields \(x=-9\). Thus, the \(x\)-intercepts are (-1, 0) and (-9, 0).

Step 4: Find the y-intercept

Set \(x=0\) and solve the equation for \(y\), which yields \(y = 0^{2} +10*0+9=9\). Thus, the y-intercept is (0, 9).

Step 5: Graph the quadratic function

Plot the vertex (-5,-16), the y-intercept (0, 9) and x-intercepts (-1, 0) and (-9, 0). Reflect points along the axis of vertical symmetry, x=-5. Connect points with a smooth curve opening upwards to complete the graph of the quadratic function.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is an essential point that can tell a lot about the graph of a quadratic function. For a quadratic equation written in the form \(y = ax^2 + bx + c\), the vertex can be found using the formula:

• \(h = -\frac{b}{2a}\)

Once you have \(h\), plug it back into the equation to find \(k\) using:

• \(k = ah^2 + bh + c\)

This gives you the vertex \((h, k)\), a point where the parabola changes its direction.
In our example, the vertex is found at \((-5, -16)\). Since the coefficient \(a\) is greater than zero, this vertex is the lowest point on the parabola. The vertex helps us understand the "U" shape of the parabola.
It also divides the parabola into two symmetrical halves, making it easier to graph.

X-Intercepts

X-intercepts are the points where the parabola crosses the x-axis. At these points, the value of \(y\) is zero. It involves solving the equation \(ax^2 + bx + c = 0\) for \(x\).
In the given problem, the quadratic equation is \(x^2 + 10x + 9\). To determine these intercepts:

• Factor the equation to get \((x + 1)(x + 9) = 0\).
• Set each factor equal to zero and solve for \(x\).

This yields two solutions: \(x = -1\) and \(x = -9\). Thus, the x-intercepts are the points \((-1, 0)\) and \((-9, 0)\).
These points assist in plotting the shape of the parabola and are crucial for sketching an accurate graph.

Y-Intercept

The y-intercept of a quadratic function is the point where the parabola crosses the y-axis. Unlike x-intercepts, there's only one y-intercept unless the graph is a straight line.
To find the y-intercept, simply set \(x\) to zero in the quadratic equation and solve for \(y\).

• For our equation \(y = x^2 + 10x + 9\), setting \(x = 0\) gives \(y = 9\).

Hence, the y-intercept is \((0, 9)\).
This point is important because it provides a reference point for the graph and indicates the constant term \(c\) from the equation. It also helps in confirming other points plotted on the graph.

Graphing Quadratic Functions

Graphing a quadratic function combines all the elements we've discussed: the vertex, x-intercepts, and y-intercepts. These are the main features that create the parabola shape.
Start by plotting the vertex, as it indicates the axis of symmetry and the "turning point" of the graph. Next, mark the x-intercepts, which define the points where the parabola crosses the x-axis.
Then, plot the y-intercept. This ensemble of points provides the framework of the parabola.
For continuity of the drawn curve and for more accuracy:

• Include other points, using symmetry about the axis of the vertex.
• Interpolate smoothly between plotted points.

In our task, since the parabola opens upwards, connect the points with a smooth "U-shaped" curve. Always check your graph for symmetry and ensure it includes all key features.
Graphing aids in visualizing how quadratic expressions behave and is a valuable skill in mathematics.