Problem 17 - Algebra and Trigonometry

Question

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=1-x^{2} $$

Step-by-Step Solution

Answer

The x-intercepts are \((-1, 0)\) and \((1, 0)\), and the y-intercept is \((0, 1)\).

Step 1: Create Table of Values

To create a table of values, choose several values for \(x\) and then find the corresponding \(y\)-values using the equation \(y = 1 - x^2\). For example, choose \(x\) values such as -2, -1, 0, 1, and 2 to get:- For \(x = -2\), \(y = 1 - (-2)^2 = 1 - 4 = -3\)- For \(x = -1\), \(y = 1 - (-1)^2 = 1 - 1 = 0\)- For \(x = 0\), \(y = 1 - 0^2 = 1\)- For \(x = 1\), \(y = 1 - 1^2 = 0\)- For \(x = 2\), \(y = 1 - 2^2 = 1 - 4 = -3\)The table of values is: \(x: [-2, -1, 0, 1, 2]\)\(y: [-3, 0, 1, 0, -3]\)

Step 2: Sketch the Graph

Using the table of values, plot the points \((-2,-3)\), \((-1,0)\), \((0,1)\), \((1,0)\), and \((2,-3)\) on a coordinate plane. Connect the points with a smooth curve to form a parabola opening downwards. This graph represents the equation \(y = 1 - x^2\).

Step 3: Find the y-intercept

The \(y\)-intercept is where the graph crosses the \(y\)-axis, which occurs when \(x = 0\). Substitute \(x = 0\) into the equation to find \(y\):\(y = 1 - 0^2 = 1\)Therefore, the \(y\)-intercept is at \((0, 1)\).

Step 4: Find the x-intercepts

The \(x\)-intercepts occur where the graph crosses the \(x\)-axis, meaning \(y = 0\). Set \(y = 0\) in the equation:\(0 = 1 - x^2\)Rearrange and solve the equation:\(x^2 = 1\)\(x = \pm\sqrt{1}\)\(x = \pm 1\)The \(x\)-intercepts are \((-1, 0)\) and \((1, 0)\).

Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards or downwards. It's the graphical representation of a quadratic equation, which is an equation of the form \(y = ax^2 + bx + c\).
The parabola's vertex, where it changes direction, depends on the coefficients \(a, b, \) and \(c\).
In the equation \(y = 1 - x^2\), the parabola opens downwards because the coefficient of \(x^2\) is negative.

• Vertex: The highest or lowest point on the parabola, found by using the formula \(x = -\frac{b}{2a}\).
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror images.
• Direction: A parabola that has a negative leading coefficient opens downwards, while a positive one opens upwards.

Understanding the structure of a parabola helps in predicting how it will look on the graph and what its specific characteristics will be.

Graphing

Graphing a quadratic equation involves plotting points and sketching the curve based on the function's values. It's a visual representation of the relationships expressed in the equation, such as \(y = 1 - x^2\). This forms a parabola.
Here's how to graph it:

• Start by creating a table of values, selecting different \(x\) values to find corresponding \(y\) values.
• Plot these points on a graph with the \(x\) and \(y\) axes marked.
• Connect the plotted points smoothly to show the parabola shape.

Since our example has no linear term (the \(bx\) part is zero), the vertex is straightforward to find at \(x = 0\), making it easy to graph. Ensure your plot is smooth and curves, reflecting the continuous nature of a quadratic function.
Accuracy in graphing helps in clearly seeing the intercepts and the overall path the parabola takes.

Intercepts

Intercepts are points where the graph intersects the axes, providing crucial information about the equation.
In quadratic equations like \(y = 1 - x^2\), both \(x\)- and \(y\)-intercepts are important:

• Y-intercept: This is the point where the graph crosses the \(y\)-axis, found by setting \(x = 0\) in the equation. For \(y = 1 - x^2\), the y-intercept is \((0, 1)\), showing where the parabola reaches the \(y\)-axis.
• X-intercepts: These are the points where the graph crosses the \(x\)-axis, determined by setting \(y = 0\) and solving the equation. In this case, \(1 - x^2 = 0\) leads to \(x = \pm1\). Thus, the x-intercepts are \((-1, 0)\) and \((1, 0)\).

Finding these intercepts helps in understanding where the curve lies in relation to the axes and provides points of reference for graphing. These points are often symmetrical about the vertex, reflecting the parabola's axis of symmetry. Identifying intercepts and plotting them correctly ensures the accuracy of your graph.