Question

Determine whether the graph of the functions $f\left(x\right)=2x^2+5x-2$and $f\left(x\right)=-3x^2-4x+7$open downward or upward.

Short answer

The graph of the function$f\left(x\right)=2x^2+5x-2$opens upward whereas the graph of$f\left(x\right)=-3x^2-4x+7$opens downward.

Step-by-step solution

Part (a) Step 1 Given information

The function is given as $f\left(x\right)=2x^2+5x-2$to find the graph opening directions.

Find the direction

Apply the properties of the quadratic functions to find the required direction.

• If the quadratic equation is given as $f\left(x\right)=a{x}^2+bx+c$, where a is positive, the graph of the function opens upward.
• Comparison of the given equation with the standard quadratic equation results in the value of a, which is 2.
• Hence the graph of the function opens upward.

Part (b) Step 1 Given information

The function is given as$f\left(x\right)=-3x^2-4x+7$to find the opening direction.

Find the direction

Apply the properties of the quadratic functions to find the required direction.

• If the quadratic equation is given as $f\left(x\right)=a{x}^2+bx+c$, where a is negative, the graph of the function opens downward.
• Comparison of the given equation with the standard quadratic equation results in the value of a, which is -3.
• Hence the graph of the function opens downward.