Question

determine the characteristics of the quadratic function shown below.
a. the direction in which the graph opens
b. the axis of symmetry
c. the turning point (vertex)
d. the y-intercept
y = 3x² - 3x - 7

a. the parabola opens downward, upward

Explanation:

Step1: Recall the rule for parabola direction

For a quadratic function \( y = ax^2 + bx + c \), if \( a>0 \), the parabola opens upward; if \( a<0 \), it opens downward.

Step2: Identify the value of \( a \)

In the function \( y = 3x^2 - 3x - 7 \), the coefficient of \( x^2 \) (i.e., \( a \)) is \( 3 \). Since \( 3>0 \), the parabola opens upward.

Step3: Axis of symmetry formula

The formula for the axis of symmetry of a quadratic function \( y = ax^2 + bx + c \) is \( x = -\frac{b}{2a} \). Here, \( a = 3 \) and \( b = -3 \). Substituting these values: \( x = -\frac{-3}{2\times3}=\frac{3}{6}=\frac{1}{2} \).

Step4: Find the vertex (turning point)

The x - coordinate of the vertex is the axis of symmetry, \( x=\frac{1}{2} \). To find the y - coordinate, substitute \( x = \frac{1}{2} \) into the function:
\( y=3\times(\frac{1}{2})^2-3\times\frac{1}{2}-7=3\times\frac{1}{4}-\frac{3}{2}-7=\frac{3}{4}-\frac{6}{4}-\frac{28}{4}=\frac{3 - 6 - 28}{4}=\frac{-31}{4}=-7.75 \). So the vertex is \( (\frac{1}{2},-\frac{31}{4}) \).

Step5: Find the y - intercept

The y - intercept of a function \( y = ax^2+bx + c \) is the value of \( y \) when \( x = 0 \). Substitute \( x = 0 \) into \( y = 3x^2-3x - 7 \): \( y=3\times0^2-3\times0 - 7=-7 \).

Answer:

s:
a. upward
b. \( x=\frac{1}{2} \)
c. \( (\frac{1}{2},-\frac{31}{4}) \) (or \( (\frac{1}{2}, - 7.75) \))
d. \( -7 \)