Question

this is a graph of which function?
a. (f(x) = 4x^2 - 5)
b. (f(x) = x^2 - 1)
c. (f(x) = -3x^2 + 8)
d. (f(x) = -.2x^2 + 5)
e. (f(x) = 0.1x^2 + 1)

Explanation:

Identify the vertex of the parabola

We observe the given graph to find the vertex of the parabola.
Using the Parabola Vertex Identification knowledge point:
The vertex is the lowest point of this upward-opening parabola.
Looking at the grid, the vertex lies on the y-axis at \( (0, 1) \).
This means the y-intercept of the function is \( 1 \).

Determine the direction of opening

We analyze the orientation of the parabola.
Using the Parabola Direction knowledge point:
The parabola opens upwards, which means the leading coefficient \( a \) must be positive (\( a > 0 \)).
This eliminates options with a negative leading coefficient.

Evaluate the vertex form of the quadratic function

We write the general equation for a quadratic function with vertex \( (h, k) \).
Using the Quadratic Functions knowledge point:
The vertex form is:

$$ f(x) = a(x - h)^2 + k $$

Substituting the vertex \( (0, 1) \):

$$ f(x) = ax^2 + 1 $$

This matches the structure of option (e), where \( a = 0.1 \) and \( k = 1 \).

Verify with a point on the graph

We check if the function \( f(x) = 0.1x^2 + 1 \) matches the width of the parabola.
The parabola is very wide, indicating a small positive leading coefficient.
If \( x = 5 \):

$$ f(5) = 0.1(5)^2 + 1 = 0.1(25) + 1 = 3.5 $$

Looking at the graph, at \( x = 5 \), the curve is indeed between \( y = 3 \) and \( y = 4 \), confirming the function.

Answer:

• a. \(f(x) = 4x^2 - 5\)
• b. \(f(x) = x^2 - 1\)
• c. \(f(x) = -3x^2 + 8\)
• d. \(f(x) = -.2x^2 + 5\)
• e. \(f(x) = 0.1x^2 + 1\) (Correct answer)