Questions: Test 2 (Covers Sections: 2.3, 2.4, 2.5, 3.2, 3.3) Question 26 of 26 For the function below, (a) find the vertex; (b) find the axis of symmetry; (c) determine whether there is a maximum or a minimum value and find that value; and (d) graph the function. f(x)=2x^2+10x+15 (a) The vertex is (Type an ordered pair, using integers or fractions.) (b) The axis of symmetry is (Type an equation. Use integers or fractions for any numbers in the equation.) (c) Does f(x) have a maximum or a minimum value?

Solution Steps

The vertex of a quadratic function \( f(x) = ax^2 + bx + c \) is given by the formula: \[ x = -\frac{b}{2a} \] For the function \( f(x) = 2x^2 + 10x + 15 \), we have \( a = 2 \) and \( b = 10 \).

Calculate \( x \): \[ x = -\frac{10}{2 \times 2} = -\frac{10}{4} = -2.5 \]

Substitute \( x = -2.5 \) back into the function to find \( y \): \[ f(-2.5) = 2(-2.5)^2 + 10(-2.5) + 15 = 2(6.25) - 25 + 15 = 12.5 - 25 + 15 = 2.5 \]

The vertex is \((-2.5, 2.5)\).

The axis of symmetry for a quadratic function is the vertical line that passes through the vertex. It is given by the equation: \[ x = -\frac{b}{2a} \] Thus, the axis of symmetry is: \[ x = -2.5 \]

Since the coefficient of \( x^2 \) (which is 2) is positive, the parabola opens upwards. Therefore, the function has a minimum value at the vertex.

The minimum value of \( f(x) \) is \( 2.5 \).

Final Answer