Question

Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry. $$ y=\frac{1}{2} x^{2}+3 x-7 $$

Short answer

Thus, the graph of the function \(y = 0.5x^2 + 3x - 7\) opens upwards, the coordinates of the vertex are \(-3, -4.5\), and the equation of the axis of symmetry is \(x = -3\).

Step-by-step solution

Determine the Opening of the Graph

For the given quadratic equation \(y = 0.5x^2 + 3x - 7\), the coefficient of \(x^2\) is 0.5, which is a positive number. Therefore, the graph of the given function opens upward.

Find the Coordinates of the Vertex

The coordinates of the vertex for any standard form quadratic equation can be found using the formula \((-b/2a , f(-b/2a))\). For the given equation \(y = 0.5x^2 + 3x - 7\), a = 0.5 and b = 3. Plugging this into the formula, we get the x-coordinate of the vertex as \(-3/(2*0.5) = -3\). To find the y-coordinate, substitute \(-3\) into the equation, getting \(y = 0.5(-3)^2 + 3(-3) - 7 = -4.5\). Hence, the coordinates of the vertex are \(-3, -4.5\).

Write an Equation of the Axis of Symmetry

The axis of symmetry for a standard form quadratic equation can be found using the formula \(x = -b/2a\). For the given equation \(y = 0.5x^2 + 3x - 7\), a = 0.5 and b = 3. Plugging this into the formula, we get the equation of the axis of symmetry as \(x = -3\).