Graphing
What is the shape of a quadratic graph called?
Parabola
What does it mean to factor a quadratic?
Find the two factors that multiply to equal C and add to equal B (ax2 + bx + C)
What is the quadratic formula?
-b plus or minus the square root of b2-4ac all divided by 2a
A ball is thrown upward from a height of 3 feet. The height of the ball after tt seconds is given by h(t)=−16t2+32t+3h. What is the maximum height reached by the ball?
The maximum height reached by the ball is 35 feet (calculated by finding the vertex of h(t))
What is a vertical shift in a quadratic function?
A vertical shift moves the graph up or down based on the constant added or subtracted from the function.
How do you determine if a parabola opens upward or downward?
If the equation is positive or negative
Factor the expression: x2+5x+6
(x+3) (x+2)
How do you use the quadratic formula to find the roots?
Substitute a, b, and c into the quadratic formula and solve for x.
A rectangular garden has a length that is 2 meters longer than its width. If the area of the garden is 48 square meters, find the dimensions of the garden.
The dimensions of the garden are 6 meters (width) and 8 meters (length).
How does changing the value of aa affect the graph of a quadratic function?
Changing the value of aa affects the width and direction of the parabola; if , the graph is narrower, and if , the graph is wider.
What is the axis of symmetry?
The line that cuts the parabola in half
or
-b/2a
Factor the Expression:
x2 + 2x - 24
(x + 6 ) (x - 4)
What do the discriminant values indicate about the roots of a quadratic?
If the discriminant is positive, there are two real roots; if zero, one real root; if negative, no real roots.
A company's profit PP in thousands of dollars can be modeled by the equation P(x)=−x2+20x−50, where x is the number of units sold. How many units must be sold to maximize profit?
To maximize profit, sell 10 units (calculated by finding the vertex of P(x)).
Describe a horizontal stretch or compression of a quadratic function.
A horizontal stretch or compression alters the width of the parabola; multiplying xx by a factor greater than 1 compresses it, while a factor between 0 and 1 stretches it.
How do you find the x-intercepts of a quadratic function?
Set the equation equal to 0 and solve by...
or
Desmos
Explain how to use factoring to solve a quadratic equation.
Use factoring to set each factor to zero and solve for x.
Solve for roots using the quadratic formula: 2x2+4x−6=0.
Roots are x=1 and x=−3
A car’s path is modeled by the equation h(t)=−5t2+20t+1 where h is the height in meters and tt is time in seconds. When will the car hit the ground?
The car will hit the ground after 4 seconds (solving −5t2+20t+1=0).
What is a reflection over the x-axis in the context of quadratics?
A reflection over the x-axis occurs when aa is negative, causing the parabola to flip upside down.
Explain how to graph a quadratic function step-by-step.
Type it into desmos!
How do you determine if a quadratic is factorable?
A quadratic is factorable if the discriminant b2−4ac is a perfect square.
Explain how the quadratic formula relates to graphing a quadratic function.
The quadratic formula provides the x-values where the graph intersects the x-axis.
A projectile is launched from the ground with an initial velocity of 50 m/s. The height of the projectile can be modeled by h(t)=−4.9t2+50t. How long will it take for the projectile to reach its maximum height?
The projectile will reach its maximum height in approximately 5.1 seconds (vertex of h(t)).
How do you combine transformations to graph a quadratic function?
To combine transformations, apply vertical and horizontal shifts, stretches/compressions, and reflections in the order of horizontal shifts, vertical shifts, and then stretches/compressions.