To describe the graph of the quadratic function [tex]\( y = 5x^2 - 3x + 21 \)[/tex], let's break down the problem step-by-step.
1. Determine the direction in which the parabola opens:
- The general form of a quadratic function is [tex]\( y = ax^2 + bx + c \)[/tex].
- The direction in which the parabola opens is determined by the coefficient [tex]\( a \)[/tex] of the [tex]\( x^2 \)[/tex] term.
- If [tex]\( a \)[/tex] is positive, the parabola opens upward.
- If [tex]\( a \)[/tex] is negative, the parabola opens downward.
For our function [tex]\( y = 5x^2 - 3x + 21 \)[/tex]:
- The coefficient of the [tex]\( x^2 \)[/tex] term is [tex]\( 5 \)[/tex], which is positive.
Therefore, the parabola opens upward.
2. Determine the shape of the parabola compared to [tex]\( y = x^2 \)[/tex]:
- The shape of the parabola is influenced by the value of [tex]\( a \)[/tex].
- If the absolute value of [tex]\( a \)[/tex] is greater than 1, the parabola is narrower than [tex]\( y = x^2 \)[/tex].
- If the absolute value of [tex]\( a \)[/tex] is less than 1, the parabola is wider than [tex]\( y = x^2 \)[/tex].
- If the absolute value of [tex]\( a \)[/tex] is equal to 1, the parabola has the same width as [tex]\( y = x^2 \)[/tex].
For our function [tex]\( y = 5x^2 - 3x + 21 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is [tex]\( 5 \)[/tex], which has an absolute value greater than 1.
Therefore, the parabola is narrower than [tex]\( y = x^2 \)[/tex].
So, filled in the blanks, we get:
The parabola opens [tex]\(\boxed{\text{upward}}\)[/tex], and its shape is [tex]\(\boxed{\text{narrower}}\)[/tex] than [tex]\( y = x^2 \)[/tex].
