Please explain to me how to do the following problems, I don't understand them. Thanks!
Find the x-intercepts (if any) for the graph of the quadratic function.
1.f(x) = (x + 2)2 - 4
Find the y-intercept for the graph of the quadratic function.
5.y + 9 = (x + 3)2
Find the domain and range of the quadratic function whose graph is described.
7.The vertex is (1, -15) and the graph opens up.
To find the x-intercept, set y=0 and solve for x
To find the y-intercept, set x=0 and solve for y
the domain of a standard parabola that opens either up or down is
x "is an element of" R
the domain is either y ≥ "the y value of the vertex", if it opens up or
y ≤ "the y value of the vertex" if it opens downwards
Sure! I'd be happy to explain how to solve each of these problems.
1. To find the x-intercepts of a quadratic function, you set the function equal to zero and solve for x. In this case, the given quadratic function is f(x) = (x + 2)2 - 4. To find the x-intercepts, we set f(x) = 0 and solve for x.
So, we have (x + 2)2 - 4 = 0.
Next, we can solve this equation by expanding the squared term and simplifying. Let's do the calculations step by step:
(x + 2)2 - 4 = 0
(x + 2)(x + 2) - 4 = 0
(x2 + 4x + 4) - 4 = 0
x2 + 4x = 0
x(x + 4) = 0
From this equation, we can see that the x-intercepts occur when either x = 0 or x + 4 = 0. So, the x-intercepts are x = 0 and x = -4.
2. To find the y-intercept of a quadratic function, we substitute x = 0 into the equation. In this case, the given quadratic function is y + 9 = (x + 3)2. We want to find the value of y when x = 0.
So, we substitute x = 0 into the equation:
y + 9 = (0 + 3)2
y + 9 = 32
y + 9 = 9
y = 0
Therefore, the y-intercept is at the point (0, 0).
3. To find the domain and range of a quadratic function, we need to understand what values x and y can take.
Given that the vertex is (1, -15) and the graph opens up, we know that the vertex is the lowest point along the parabola. This information provides insights into the range and domain of the quadratic function.
For the domain, since the graph opens upward and there are no restrictions mentioned, the domain is all real numbers. Therefore, the domain is (-∞, ∞).
For the range, we can conclude that the graph opens upward, which means it reaches its lowest point at the vertex. The y-coordinate of the vertex is -15. Since there is no upper limit for the y-values, the range extends from -15 to positive infinity. Therefore, the range is (-15, ∞).
I hope this helps! Let me know if you have any further questions.