To solve these questions, we will need to first analyze the given quadratic functions and their properties. The general form of a quadratic function is:

$f(x) = ax^2 + bx + c$

Function 1:

$f(x) = 2x^2 - 8x + 15$

(a) Vertex of Function 1:

For a quadratic function $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex is given by the formula:

$x_{\text{vertex}} = \frac{-b}{2a}$

For Function 1:

• $a = 2$
• $b = -8$

So,

$x_{\text{vertex}} = \frac{-(-8)}{2(2)} = \frac{8}{4} = 2$

Now substitute $x = 2$ into the equation to find the y-coordinate of the vertex:

$f(2) = 2(2)^2 - 8(2) + 15 = 2(4) - 16 + 15 = 8 - 16 + 15 = 7$

Thus, the vertex of Function 1 is $(2, 7)$.

(b) Vertex of Function 2:

The data points for Function 2 are not directly in equation form, but based on the points provided, we can identify the vertex. Observing the table, it seems that the function has a minimum value at $x = -3$ (where the value of $y$ is the lowest, 1). Hence, we deduce the vertex of Function 2 to be $(-3, 1)$.

(c) Which function has the smaller minimum value?

To determine which function has the smaller minimum value, we compare the y-coordinates of the vertices.

• Function 1's minimum value is 7 (from the vertex $(2, 7)$).
• Function 2's minimum value is 1 (from the vertex $(-3, 1)$).

Thus, Function 2 has the smaller minimum value, which is 1.

Summary:

• (a) The vertex of Function 1 is $(2, 7)$.
• (b) The vertex of Function 2 is $(-3, 1)$.
• (c) Function 2 has the smaller minimum value, and the smaller minimum value is 1.

Would you like any further details or clarifications on these answers?

Here are 5 related questions for further exploration:

1. How do you determine the direction of opening (upwards or downwards) of a quadratic function from its equation?
2. How can you calculate the axis of symmetry for a quadratic function?
3. What is the significance of the vertex in relation to the graph of a quadratic function?
4. How do you convert the standard form of a quadratic function to vertex form?
5. Can you explain how to find the roots or x-intercepts of a quadratic function?

Tip: If you have trouble remembering the vertex formula, just remember that the x-coordinate of the vertex always occurs at $\frac{-b}{2a}$ for a quadratic equation in standard form!