Problem 14 - 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. \(y=-x^{2}+2 x+3\)

Step-by-Step Solution

Answer

a. The parabola opens downward. b. The vertex is at (1, 4). c. The x-intercepts are x = -1 and x = 3. d. The y-intercept is y = 3. e. Graph the function using the vertex and the intercept points, remembering that the graph opens downward.

Step 1: Determine the direction of parabola

The coefficient of the \(x^{2}\) term of the given quadratic equation \(y=-x^{2}+2 x+3\) is negative (-1). Therefore, the parabola opens downward.

Step 2: Find the vertex of the parabola

The vertex form of a parabola is given as \(y=a(x-h)^2+k\), where (h,k) are the coordinates of the vertex. The h-value can be found using the formula \(h=-\frac{b}{2a}\). For the given equation, a = -1 and b = 2. Plugging these values into the formula, we get \(h=-\frac{2}{2(-1)}=1\). Now substitute x=1 into the equation to get y value. \(y=-1*(1)^2+2*(1)+3=4\). Hence, the vertex is (1,4).

Step 3: Find the x-intercepts of the parabola

The x-intercepts are the x-values where the function output is zero (y = 0). Set the equation \(y=-x^{2}+2 x+3\) equal to zero and solve for x. \(0=-x^{2}+2 x+3 => x^{2}-2x-3=0\). By factoring, we get \((x-3)(x+1)=0\). So, the x-intercepts are x = 3 and x = -1.

Step 4: Find the y-intercept of the parabola

The y-intercept is where the parabola crosses the y-axis (x = 0). Substitute x=0 into the equation: y = -0+0+3, thus the y-intercept is y=3.

Step 5: Sketch the graph using the obtained information

Using the vertex at (1,4), the x-intercepts at (-1,0) and (3,0), the y-intercept at (0,3) and the knowledge that the parabola opens downward, hypothesize and sketch the graph of the quadratic function.

Key Concepts

Parabola Direction

Understanding the direction in which a parabola opens is crucial when graphing quadratic functions. The direction is determined by the coefficient of the quadratic term, which is the term with the variable raised to the second power. In the case of the quadratic equation y = -x^2 + 2x + 3, the coefficient of the x^2 term is negative (-1). This negative value tells us that the parabola opens downward.

When a parabola opens downward, its arms extend down from the vertex, resembling an upside-down 'U'. A positive coefficient would indicate that the parabola opens upward, similar to a right-side-up 'U'. Recognizing this characteristic is the first step in sketching the overall shape of the parabola on a graph.

Vertex of a Parabola

The vertex of a parabola is the peak or the lowest point, depending on whether the parabola opens upward or downward. It's one of the most important features to identify because it marks the turning point of the graph. For the quadratic function in question, y = -x^2 + 2x + 3, we can find the vertex by using the formula h = -b/(2a).

With our given values of a = -1 and b = 2, the calculation of h gives us the x-coordinate of the vertex. After substituting it into the original equation, we determine the y-coordinate, resulting in the vertex (1,4). Understanding how to find and interpret the vertex allows us to plot the most significant point of the parabola.

X-Intercepts

The x-intercepts of a parabola, also known as zeros or roots, are the points where the graph crosses the x-axis. These are found by setting the output (y) to zero and solving for x. When we solve the equation 0 = -x^2 + 2x + 3, we identify the x-intercepts to be x = 3 and x = -1.

Knowing the x-intercepts is essential for sketching a precise graph of a quadratic function. They provide clear points that you can plot and are especially helpful for determining the width and position of the parabola on the graph. Identifying these intercepts is often accomplished through factoring or applying the quadratic formula if the equation cannot be factored easily.

Y-Intercept

Just like the x-intercepts, finding the y-intercept of a parabola is a critical step in graphing a quadratic function. The y-intercept is the point where the graph crosses the y-axis, and to find it, we simply set x to zero in the original equation. Applying this to our equation yields y = 3 as the y-intercept.

The y-intercept gives us another clear point on the graph and helps us understand where the parabola meets the y-axis. This information, along with the direction, vertex, and x-intercepts, provides us with all the necessary points to sketch an accurate graph of the quadratic function.