On graph paper, use a table to make a complete graph of y = x2 − 2x − 8. Homework Help ✎
Write the coordinates of the y-intercept. What is the connection between the y‑intercept and the equation y = x2 − 2x − 8?
Write the coordinates of the x-intercepts.
What is the lowest point on the graph?
To make a complete graph of the equation y = x^2 - 2x - 8, let's start by creating a table.
We can choose some x-values and calculate the corresponding y-values using the equation.
Let's choose x-values from -5 to 5. Substitute each x-value into the equation and solve for y.
x | y
---------
-5 |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
5 |
To find the y-intercept, substitute x = 0 into the equation y = x^2 - 2x - 8:
y = (0)^2 - 2(0) - 8
Simplifying, we find that the y-intercept is at (0, -8).
The y-intercept is the point where the graph intersects the y-axis, and it is the value of y when x = 0.
To find the x-intercepts, substitute y = 0 into the equation y = x^2 - 2x - 8:
0 = x^2 - 2x - 8
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. After solving, we find the x-intercepts to be (-2, 0) and (4, 0).
The x-intercepts are the points where the graph intersects the x-axis, and they are the values of x when y = 0.
To find the lowest point on the graph, we need to find the vertex of the parabola. The vertex represents the lowest point of a quadratic function in standard form.
The vertex of the equation y = x^2 - 2x - 8 can be found using the formula: x = -b/2a.
Here, a = 1 and b = -2.
x = -(-2)/(2(1))
= 2
Substituting x = 2 back into the equation, we find:
y = (2)^2 - 2(2) - 8
= 4 - 4 - 8
= -8
Therefore, the lowest point on the graph is at (2, -8).
I hope this helps! Let me know if you have any further questions.
To create a complete graph of the equation y = x^2 - 2x - 8, you can use a table and plot points. Here's how you can do it:
Step 1: Create a table with columns for x and y. Choose a range of x-values to plug into the equation. For example, you can choose x-values from -5 to 5.
Step 2: Plug each x-value into the equation y = x^2 - 2x - 8 to find the corresponding y-value. Calculate and fill in the y-column of the table.
Step 3: Plot the points with x-values and their corresponding y-values on the graph paper. Connect the points with a smooth curve to give you the complete graph of the equation.
Now, let's answer the specific questions:
1. The y-intercept represents the point where the graph intersects the y-axis. To find it, you can set x = 0 in the equation y = x^2 - 2x - 8 and solve for y. Substituting x = 0 into the equation gives us y = (0)^2 - 2(0) - 8, which simplifies to y = -8. Therefore, the coordinates of the y-intercept are (0, -8).
2. The y-intercept represents the point when x is equal to 0. In other words, it's the value of y when x = 0. Looking at the equation y = x^2 - 2x - 8, we see that the y-intercept is a specific solution when x is 0. So, the y-intercept and the equation y = x^2 - 2x - 8 are connected through this specific value of x.
3. To find the x-intercepts, set y = 0 in the equation y = x^2 - 2x - 8 and solve for x. So, we have 0 = x^2 - 2x - 8. This equation can be factored as (x - 4)(x + 2) = 0. Setting each factor equal to zero gives us two possible values for x: x - 4 = 0 and x + 2 = 0. Solving these equations, we find x = 4 and x = -2. Therefore, the coordinates of the x-intercepts are (4, 0) and (-2, 0), respectively.
4. The lowest point on the graph represents the vertex of the quadratic function. To find it, you can use the formula x = -b/(2a), where a and b are the coefficients of the x^2 and x terms, respectively. In this equation, a = 1 and b = -2. Substituting those values gives x = -(-2) / (2 * 1), which simplifies to x = 1. To find the y-coordinate of the vertex, substitute x = 1 into the equation y = x^2 - 2x - 8. Doing this gives us y = (1)^2 - 2(1) - 8, which simplifies to y = -9. Therefore, the coordinates of the lowest point on the graph are (1, -9).
Of course, it is not possible to make a "complete" graph of a parabola on finite graph paper.
If y = x^2-2x-8 then clearly the y-intercept is -8, since that's what you get when x=0
x^2-2x-8 = (x-1)^2 - 9
So the vertex is at (1,-9), which is the lowest point on the graph.
For the x-intercepts, y=0, so
x-1 = ±3, so x = -2 or 4