Question

Describe the set of values less than or equal to those represented by a quadratic function with vertex (–10, –25) and containing the point (−16, negative 17 and fourth-fifths).

Answer 1

The values less than or equal to those represented by the quadratic function with a vertex at (-10, -25) are all the values on the function from the vertex to any given point. Since the parabola opens upwards, these will include all y-values from -25 up to the y-value corresponding to the given x-value on the function.

Step-by-step explanation:

The student is asking about the set of values that a quadratic function takes on, especially those less than or equal to a certain point. The quadratic function has a vertex at (–10, –25) and it passes through the point (–16, negative 17 and four-fifths). To find the set of values less than or equal to those represented by the function, it is important to understand the nature of a quadratic graph. A quadratic function is typically of the form ax² + bx + c, and its graph is a parabola.

If the coefficient 'a' in the quadratic function is positive, the parabola opens upwards, and the vertex represents the minimum point. Since the function in question has a vertex with a negative y-value, we can infer that the parabola opens upwards, and therefore, all values on the function are greater than or equal to the y-value of the vertex. Thus, the set of values less than or equal to any given point on the function will include all the values from the vertex up to that point on the function. In this case, all values are less than or equal to –25, and we can determine specific values by substituting corresponding x-values into the function.

Exemplifying with the point given (–16, –17.8), since it lies on the function, all x-values from –10 to –16 will yield y-values less than or equal to –17.8. Therefore, this region of the graph to the left of the vertex is what the student should consider when describing the set of values less than or equal to those represented by the quadratic function.

Answer 2

For y ≤
`(1/5)(x + 10)^2` - 25, the set of values is y ≤ -25.

For y ≥
`(1/5)(x + 25)^2` - 10, the set of values is y ≥ -10.

For y ≤
`(1/5)(x + 10)^2` - 25, the set of values is y ≤ -25.

For y <
`(1/5)(x - 10)^2` - 25, the set of values is y < 110.2.

To describe the set of values less than or equal to those represented by a quadratic function with vertex (-10, -25) and containing the point (-16, -17/5), we need to analyze the quadratic function and find the range of values for which the inequalities hold.

The quadratic function can be expressed as:

f(x) =
`(1/5)(x + 10)^2` - 25

Let's analyze each inequality step by step:

1. y ≤
`(1/5)(x + 10)^2` - 25:

To find the set of values that satisfy this inequality, we need to determine the range of the quadratic function.

First, we find the vertex of the quadratic function, which is given as (-10, -25). This vertex is the minimum point of the parabola.

The minimum value of the quadratic function occurs at the vertex, which is -25. Therefore, all values of y less than or equal to -25 satisfy this inequality.

So, the set of values that satisfy y ≤
`(1/5)(x + 10)^2` - 25 is y ≤ -25.

2. y ≥
`(1/5)(x + 25)^2` - 10:

To find the set of values that satisfy this inequality, we again need to determine the range of the quadratic function.

The vertex of this quadratic function is given as (-25, -10), which is also the minimum point of the parabola.

The minimum value of the quadratic function occurs at the vertex, which is -10. Therefore, all values of y greater than or equal to -10 satisfy this inequality.

So, the set of values that satisfy y ≥
`(1/5)(x + 25)^2` - 10 is y ≥ -10.

3. y ≤
`(1/5)(x + 10)^2` - 25:

This inequality is similar to the first one we analyzed, and the set of values that satisfy it is y ≤ -25, as we found earlier.

4. y <
`(1/5)(x - 10)^2` - 25:

To find the set of values that satisfy this inequality, we need to determine the range of the quadratic function with a vertex at (-10, -25) and containing the point (-16, -17/5).

Let's analyze this case separately:

• The vertex of the quadratic function is (-10, -25), which is the minimum point of the parabola.
• The point (-16, -17/5) lies on the parabola.

To find the range of values satisfying y < (1/5)(x - 10)^2 - 25, we can compare it with the vertex and the given point.

Substituting x = -16 into the quadratic function:

• y =
  `(1/5)(-16 - 10)^2` - 25
• y =
  `(1/5)(-26)^2` - 25
• y = (1/5)(676) - 25
• y = 135.2 - 25
• y = 110.2

So, the point (-16, -17/5) lies on the parabola, and its y-coordinate is approximately 110.2.

Therefore, the set of values satisfying y <
`(1/5)(x - 10)^2` - 25 is y < 110.2.

The complete question is here:

Describe the set of values less than or equal to those represented by a quadratic function with vertex (-10,-25) and containing the point (-16,-17(4)/(5)) y<=
`(1)/(5)(x+25)^{(2)`-10 y>=
`(1)/(5)(x+25)^{(2)`-10 y<=
`(1)/(5)(x+10)^{(2)`-25 y<
`(1)/(5)(x-10)^{(2)`-25.