4.06 Discriminant and parabolas
For the following graphs of functions of the form y = a x^{2} + b x + c :
State whether the vertex of the parabola is a maximum or minimum point.
State whether the value of a is negative or positive.
State the number of solutions to the equation a x^{2} + b x + c = 0.
Below is the result after using the quadratic formula to solve an equation:
x = \dfrac{- \left( - 10 \right) \pm \sqrt{ - 1 }}{8}
What can be concluded about the solutions of the equation?
By inspection, determine the number of real solutions for each of the following quadratic equations:
When graphing a particular parabola, Katrina used the quadratic formula and found that b^{2} - 4 a c = - 5. How many x-intercepts does the parabola have?
When graphing a particular parabola, Tony used the quadratic formula and found that \\ b^{2} - 4 a c = 0. How many x-intercepts does the parabola have?
For each of the following graphs of a quadratic f \left( x \right) = a x^{2} + b x + c, with discriminant \\ \Delta = b^{2} - 4ac :
State whether a \gt 0 or a \lt 0.
State whether \Delta \gt 0, \Delta \lt 0 or \Delta = 0.
Consider the graph of the quadratic function:y = m - 9 x - 3 x^{2}
Find the possible values of m, if the graph has no x-intercepts.
State the largest possible integer value of m.
Determine the value(s) of k for which the graph of y = 4 x^{2} - 4 x + k - 15 just touches the \\ x-axis.
Consider the parabola y = x^{2} - 8 x and the line y = - 7.
Form an equation to solve for the x-coordinate(s) of their point(s) of intersection.
Use the discriminant to determine how many points of intersection they have.
Find their point(s) of intersection.
Consider the parabola y = 6 x^{2} - 18 x + 51 and the line y = 18 x + 3.
Use the discriminant to determine how many points of intersection they have.
Find their point(s) of intersection.
Show that the line y = - 13 x - 10 is a tangent to the parabola y = \left(x - 3\right) \left(x - 2\right).
Consider the parabola y = - x^{2} - 3 x + 1 and the line y = - 3 x - 3.
Determine how many points of intersection they have.
Find their point(s) of intersection.
Consider the parabola y = x \left(x - 4\right) and the line y = - 3 x -5.
Form an equation to solve for the x-coordinate(s) of their point(s) of intersection.
Determine how many points of intersection they have.
Is the line y=5x-7 and tangent to the parabola y=4x^2-8x+10? Explain your answer.