Problem 13 - 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. \(y=-x^{2}+4 x-3\)

Step-by-Step Solution

Answer

The parabola opens downward. The vertex is at (2,1). The \(x\)-intercepts are located at (1,0) and (3,0), and the \(y\)-intercept is at (0,-3). These points can be used to graph the function.

Step 1: Direction of Parabola

Analyze the coefficient of \(x^{2}\). If it is positive, the parabola opens upward. If it is negative, the parabola opens downward. For \(y=-x^{2}+4x-3\), the coefficient of \(x^{2}\) is -1. So, the parabola opens downward.

Step 2: Finding the Vertex

The vertex \((h, k)\) can be found using the equations \(h = -b/2a\) and \(k = c - b^{2}/4a\). Here, \(a=-1\), \(b=4\), and \(c=-3\). Therefore, \(h = -b/2a = -4/(-2) = 2\) and \(k = c - b^{2}/4a = -3 - (4^{2}/-4) = -3 - (-4) = 1\). So, the vertex is (2,1).

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

To find the \(x\)-intercepts, set \(y=0\) in the equation and solve for \(x\). So, \(0 = -x^{2} + 4x - 3\). Solving this equation gives \(x = 1\) and \(x = 3\). Therefore, the \(x\)-intercepts are (1,0) and (3,0).

Step 4: Finding the \(y\)-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation and solve for \(y\). Therefore, \(y = -(0)^{2} + 4*0 - 3 = -3\). So, the \(y\)-intercept is (0,-3).

Step 5: Graphing the Function

The function can now be graphed. Mark the \(x\)- and \(y\)- intercepts, and the vertex on the x-y plane. Because the parabola opens downward, mark an additional point to the right of the vertex and mirror it to the left of the vertex. Draw a smooth curve through these points to complete the graph.

Key Concepts

Parabola Orientation

Understanding the direction in which a parabola opens is fundamental when dealing with quadratic functions. The term parabola orientation refers to whether the parabola faces upwards or downwards. This feature is dictated by the sign of the coefficient of the quadratic term - that is, the coefficient of the x² term.

In the example y=-x²+4x-3, the coefficient before x² is -1. This negative sign tells us that the parabola opens downward. A parabola that opens downward has a maximum point which is called the vertex. On the contrary, if the coefficient had been positive, the parabola would open upward, indicating that the vertex would be at the minimum point of the parabola.

To visualize this, imagine an umbrella. If the coefficient is negative, the umbrella is upside down, collecting rain (downward orientation). If positive, the umbrella is right-side-up, shedding rain off its sides (upward orientation).

Vertex of a Parabola

The vertex is a crucial point on a parabola, marking its peak if the parabola opens downwards, or its lowest point if it opens upwards. Mathematically, for a quadratic function in the form y = ax² + bx + c, the vertex (h, k) can be calculated using the formula h = -b/(2a) and k = c - (b²)/(4a).

For the given function, with a=-1, b=4, and c=-3, we find that h, which is the x-coordinate of the vertex, equals 2. The k, being the y-coordinate, equals 1. Therefore, the vertex for this quadratic function is (2, 1). The vertex provides pivotal information as it not only helps to graph the parabola accurately but also gives insight into the function's maximum or minimum value.

X-Intercepts of a Quadratic

The points where a graph intersects the x-axis are known as the x-intercepts or zeroes of the function. To find these, we need to set y to zero and solve the quadratic equation for x. This illustrates where the function's output value is zero.

In our quadratic equation y=-x²+4x-3, setting y equal to zero yields the equation 0=-x²+4x-3. Factoring or using the quadratic formula gives us two x-intercepts: x = 1 and x = 3. Thus, the graph of the function crosses the x-axis at the points (1, 0) and (3, 0). Knowing the x-intercepts is essential for sketching the graph and for understanding the roots of the quadratic equation.

Y-Intercept of a Quadratic

The y-intercept is where the parabola crosses the y-axis, and it holds significant importance because it is the output value of the function when x equals zero. To find the y-intercept, we substitute zero for x in the quadratic equation and solve for y.

In our case, substituting x with zero in y=-x²+4x-3 reveals that y is -3. Therefore, the y-intercept is located at the coordinate (0, -3). This provides us with another point through which the parabola will pass, helping further to sketch the graph of the quadratic function accurately.