Which answer choice describes how the graph of `f(x)=x^{2}` was transformed to create the graph of `h(x)=x^{2}-1`?
The graph of `h(x)=x^{2}-1` was shifted down one unit from the graph of `f(x)=x^{2}`.
Which value of `x` is a solution to this equation?
`2x^{2}+6x-20=0`
Using factoring, we can begin by factoring the equation as:
2x² + 6x - 20 = 0
2(x² + 3x - 10) = 0
2(x + 5)(x - 2) = 0
Now we can use the zero product property and set each factor equal to zero:
x + 5 = 0 or x - 2 = 0
Solving for x, we have:
x = -5 or x = 2
Therefore, the solutions to the equation `2x^{2}+6x-20=0` are x = -5 and x = 2.
1) Solve `x^{2}-4x-5=0` and enter solutions below.
2) Push the "Graph It" Button to see a graph of `y=x^{2}-4x-5`
1) First, we can use factoring to rewrite the equation as:
x² - 4x - 5 = 0
(x - 5)(x + 1) = 0
Using the zero product property, we can set each factor equal to zero:
x - 5 = 0 or x + 1 = 0
Solving for x, we get:
x = 5 or x = -1
Therefore, the solutions to the equation `x^{2}-4x-5=0` are x = 5 and x = -1.
2) (See graph below)
To determine how the graph of f(x)=x^2 was transformed to create the graph of h(x)=x^2-1, we need to examine the changes made to the original function.
The transformation from f(x) to h(x) involves subtracting 1 from the function. This means that for every x-value, h(x) will be 1 unit less than f(x).
Therefore, the correct answer choice is: The graph of h(x)=x^2-1 is the graph of f(x)=x^2 shifted downward by 1 unit.
To describe how the graph of `f(x) = x^2` was transformed to create the graph of `h(x) = x^2 - 1`, we need to understand the effect of each part of the equation.
In `f(x) = x^2`, the function represents a basic quadratic function. The graph of `f(x)` is a parabola that opens upward, centered at the origin (0, 0). It has a vertex at the point (0, 0), and the graph extends indefinitely in both positive and negative directions.
Next, in `h(x) = x^2 - 1`, we notice that a constant term `-1` is subtracted from `f(x)`. This constant term shifts the entire graph of `f(x)` downward by 1 unit. The effect of subtracting `-1` from `x^2` is that the vertex of the parabola is shifted down. Now, the vertex of the graph occurs at (0, -1) instead of (0, 0).
So, the correct answer choice would be: The graph of `f(x) = x^2` was shifted downward by 1 unit to create the graph of `h(x) = x^2 - 1`.