Problem 111 - Algebra and Trigonometry

Question

will help you prepare for the material covered in the first section of the next chapter. If \(y=1-x^{2}\), find the value of \(y\) that corresponds to values of \(x\) for each integer starting with \(-3\) and ending with 3 .

Step-by-Step Solution

Answer

The values of \(y\) for respective \(x\) from -3 to 3 are -8, -3, 0, 1, 0, -3, -8 in direct order.

Step 1: Identify the function

The function given in this problem is \(y=1-x^{2}\). This function defines a parabola which opens downwards. The task is to find corresponding \(y\) values for \(x\) from -3 to 3.

Step 2: Substitution for \(x=-3\)

First substitute -3 for \(x\) in the formula to find the corresponding value of \(y\). So, \(y = 1 - (-3)^{2} = 1 - 9 = -8\). Hence, when \(x = -3\), \(y = -8\).

Step 3: Substitution for \(x=-2\)

Then substitute -2 for \(x\) to find the next \(y\) value. So, \(y = 1 - (-2)^{2} = 1 - 4 = -3\). Hence, when \(x = -2\), \(y = -3\).

Step 4: Continue the substitution process

Repeat this process for \(x = -1\), \(0\), \(1\), \(2\), and \(3\). The value of \(y\) will be 0, 1, 0, -3, and -8 respectively. Thus, we have created a list of \(x\) values and their corresponding \(y\) values.

Key Concepts

Quadratic Function

A quadratic function is a type of function where the highest degree of the variable is 2. It can be written in the standard form as \( y = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants, and \(aeq0\). In this exercise, the quadratic function given is \( y = 1 - x^2 \). This type of function is also called a parabola, and it's a U-shaped curve. The function tells us how the variable \(x\) affects the variable \(y\). When graphing a quadratic function, the shape of the parabola depends on the sign and value of the coefficient \(a\). If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards. In the example \( y = 1 - x^2 \), \(a\) is -1, so the parabola opens downwards.

Vertex

The vertex of a quadratic function is a significant point because it's the peak or the lowest point of the parabola, depending on its orientation. For downward-opening parabolas like \( y = 1 - x^2 \), the vertex is at the top of the curve. The vertex can be found using the formula \( x = -\frac{b}{2a} \) in standard form equations, but since we only have \( y = 1 - x^2 \), the vertex is more easily spotted. Here, you can quickly see that the vertex occurs when \( x = 0 \), giving the maximum value of \( y = 1 \). Thus, the vertex for this problem is located at the point \((0, 1)\) which is also the highest point of the parabola.

Axis of Symmetry

The axis of symmetry is a straight line that divides the parabola into two mirror images. It's an imaginary line running vertically through the vertex. Every point on one side of this line has a matching point on the other side. For any quadratic equation \( y = ax^2 + bx + c \), the axis of symmetry is given by the formula \( x = -\frac{b}{2a} \). Since the given function \( y = 1 - x^2 \) is symmetric with its vertex at the origin, the axis of symmetry is simply \( x = 0 \). This helps confirm that the vertex really is at \((0, 1)\), showing how symmetry plays a key role in shaping the graph.

Graphing Quadratics

Graphing quadratic functions allows us to visualize how the function behaves. To graph a quadratic equation, like \( y = 1 - x^2 \), start by identifying key features like the vertex and axis of symmetry. Also, remember to compute the set of \(x\)-\(y\) pairs for a range of \(x\)-values, as shown in the solution step where \(x\) values of \(-3\) to \(3\) help plot individual points.

• Identify the vertex: \((0, 1)\).
• Draw the axis of symmetry: line \(x = 0\).
• Find additional points, such as \( (-3, -8), (-2, -3), (-1, 0), (0, 1), (1, 0), (2, -3), (3, -8) \).

Once you plot these points, connect them with a smooth curve to complete the parabola. Since the parabola opens downward, ensure that your vertex is the highest point on the curve. This gives a clear visual understanding of the behavior of the quadratic function.