1.07 Vertex formula
Consider the parabola of the form y = a x^{2} + b x + c, where a \neq 0.
Complete the following statement:
The x-coordinate of the vertex of the parabola occurs at x = ⬚. The y-coordinate of the vertex is found by substituting x = ⬚ into the parabola's equation and evaluating the function at this value of x.
What does the line x = \dfrac{- b}{2 a} represent on the parabola defined by the equation \\y = a x^{2} + b x + c (a \neq 0)?
Consider the equation y = 6 x - x^{2}.
Find the x-intercepts of the quadratic function.
Find the coordinates of the turning point.
Consider the parabola y = x^{2} - 18 x + 10.
Is the graph of the function concave up or down?
State the value of x at which the minimum value of the function occurs.
State the minimum y-value of the function.
Consider the quadratic function y = x^{2} - 6 x + 5.
Is the graph of the function concave up or down?
Find the x-intercepts.
State the the y-intercept.
Determine the axis of symmetry.
Hence or otherwise state the coordinates of the vertex of the parabola.
Consider the quadratic function y = - x^{2} - 2 x + 8.
Is the graph of the function concave up or down?
Find the x-intercepts of the function.
Find the y-intercept.
Determine the axis of symmetry.
Hence or otherwise find the vertex of the curve.
Consider the quadratic function y = - 8 - 6 x - x^{2}.
Is the graph of the function concave up or down?
Find the x-intercepts of the function.
Find the y-intercept.
Determine the axis of symmetry.
Hence or otherwise find the vertex of the curve.
Consider the function y = x^{2} + 4 x + 3.
Determine the equation of the axis of symmetry.
Hence determine the minimum value of y.
Hence state the coordinates of the vertex of the curve.
Here is the graph of another quadratic function. State the coordinates of its vertex.
Determine whether the following relationship is true between the graph of \\y = x^{2} + 4 x + 3 and the graph provided.
They have the same x-intercepts.
They share the same turning point.
They have the same concavity.
Consider the function y = \left(14 - x\right) \left(x - 6\right).
State the zeros of the function.
Find the axis of symmetry.
Is the graph of the function concave up or concave down?
Determine the maximum y-value of the function.
Find the coordinates of the vertex of y = 3 x^{2} - 6 x - 9.
Find the maximum value of y for the quadratic function y = - x^{2} + 10 x - 25.
Consider the quadratic function f \left( x \right) = - 7.5 x^{2} - 1.3 x - 1.3.
Find the x-coordinate of the vertex correct to three decimal places.
Hence, find the maximum value obtained by the function, correct to two decimal places.
Find the x-coordinate of the vertex of the parabola represented by P \left( x \right) = p x^{2} - \dfrac{1}{2} p x - q.
Consider these two parabolas, labeled P_{1} and P_{2}.
\begin{aligned}P_{1}: y &= x^{2} + 4 x + 6\\P_{2}:y &= x^{2} - 4 x + 6\end{aligned}
Determine the coordinates of the vertex for each parabola.
How far apart are the vertices of the two parabolas?
Consider the curve y = x^{2} + 6 x + 4.
Determine the axis of symmetry.
Hence, determine the minimum value of y.
When an object is thrown into the air, its height above the ground is given by the equation h = 193 + 24 s - s^{2}where s is its horizontal distance from where it was thrown.
Find the value of s, at the point when it reaches its greatest height above the ground.
Find the maximum height reached by the object.
The height at time t of a ball thrown upwards is given by the equation h = 59 + 42 t - 7 t^{2}.
How long does it take the ball to reach its maximum height?
Find the height of the ball at its highest point.