Math

Question Find the domain and range of a quadratic function with vertex at (4,0)(-4,0). Determine if the function opens upwards or downwards and if the graph intersects the x-axis.

Studdy Solution

STEP 1

Assumptions1. The vertex of the quadratic function is at (-4,0) . The quadratic function opens either upwards or downwards3. We need to find the domain and range of the function4. We need to determine whether the graph of the function intersects the x-axis or not

STEP 2

The general form of a quadratic function isf(x)=a(xh)2+kf(x) = a(x-h)^2 + kwhere (h,k) is the vertex of the function.

STEP 3

Substitute the given vertex (-,0) into the equation.
f(x)=a(x+)2+0f(x) = a(x+)^2 +0

STEP 4

The domain of a quadratic function is all real numbers, because a quadratic function is defined for all real values of x. Therefore, the domain is=(,) = (-\infty, \infty)

STEP 5

If the quadratic function opens upwards (a >0), the graph will be a parabola that opens upwards with a minimum point at the vertex. Therefore, the range will be all values greater than or equal to the y-coordinate of the vertex.
If the quadratic function opens downwards (a <0), the graph will be a parabola that opens downwards with a maximum point at the vertex. Therefore, the range will be all values less than or equal to the y-coordinate of the vertex.

STEP 6

Since the y-coordinate of the vertex is0, the range for a function that opens upwards isR=[0,)R = [0, \infty)and the range for a function that opens downwards isR=(,0]R = (-\infty,0]

STEP 7

If the quadratic function opens upwards (a >0), the graph will intersect the x-axis at the vertex, because the vertex is the minimum point of the graph.
If the quadratic function opens downwards (a <0), the graph will also intersect the x-axis at the vertex, because the vertex is the maximum point of the graph.
In both cases, the graph of the function intersects the x-axis at the point (-4,0).

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord