QuestionFind x-intercepts, y-intercept, domain, range of , and graph the function.
Studdy Solution
STEP 1
Assumptions1. The function is . We are asked to find the x-intercepts, y-intercept, graph the function, and determine the domain and range in interval notation.
STEP 2
First, let's find the x-intercepts. The x-intercepts are the values of where . So, we set equal to zero and solve for .
STEP 3
This is a quadratic equation, and it can be factored to solve for .
STEP 4
Setting each factor equal to zero gives the solutions.
STEP 5
olving the second equation for gives . So, the x-intercepts are and .
STEP 6
Next, let's find the y-intercept. The y-intercept is the value of when . So, we substitute into the function.
STEP 7
olving for gives . So, the y-intercept is .
STEP 8
To graph the function, use a graphing tool and plot the function . The graph should be a parabola opening upwards with the vertex at .
STEP 9
The domain of a function is the set of all possible x-values. Since there are no restrictions on in the function , the domain is all real numbers, which in interval notation is .
STEP 10
The range of a function is the set of all possible y-values. Since the graph is a parabola opening upwards, the minimum y-value is the y-coordinate of the vertex. The vertex of the parabola is at . Substituting into the function gives the minimum y-value.
STEP 11
olving for gives . So, the range of the function is .
In conclusion, the x-intercepts are and , the y-intercept is , the domain is , and the range is .
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