Quadratic Graph Practice Problems

A quadratic graph is the graph of a quadratic function of the form y = ax² + bx + c (where a ≠ 0). It is a U-shaped curve called a parabola that opens upward if a is positive and downward if a is negative.

Solved Examples

Example 1: Graph: f(x) = x² - 4x + 3

Vertex Calculation:

a = 1,
b = -4,
c = 3

Vertex x value: x = -\frac{-4}{2 \times 1} = 2
Vertex y value: f(2) = 2²- 4 **2 + 3 = -1
Vertex: (2, -1)

Axis of Symmetry: x = 2
Y-intercept: f(0) = 3
X-intercepts: Solve x² - 4x + 3 = 0
(x - 1)(x - 3) = 0
X-intercepts: x = 1 and x = 3

Graph: Plot the vertex (2, -1) the y-intercept (0, 3) and x-intercepts (1, 0) and (3, 0). Draw the parabola opening upwards.

Example 2: Graph f(x) = 2x² - 3x - 2.

Quadratic Formula:

x = \frac{b\pm \sqrt{b^{2}-4ac}}{2a}

a = 2,
b = -3,
c = -2

Discriminant: b² - 4ac = (-3)² - 4 * 2 * (-2) = 9 + 16 = 25

Roots: x = \frac{3 \pm \sqrt{25}}{4}

x = \frac{3 + 5}{4} = 2 and x = \frac{3 - 5}{4} = -\frac{1}{2}

Vertex Calculation:

Vertex x value: x = -\frac{-3}{2 \times 2} = \frac{3}{4}

Vertex y value: f\left(\frac{3}{4}\right) = 2 \left(\frac{3}{4}\right)^2 - 3 \left(\frac{3}{4}\right) - 2 = -\frac{25}{8}

Vertex: \left(\frac{3}{4}, -\frac{25}{8}\right)

Graph:Plot the roots (2, 0) and \left(-\frac{1}{2}, 0\right) and the vertex \left(\frac{3}{4}, -\frac{25}{8}\right). Draw the parabola opening upwards.

Example 3: Graph f(x) = -x² + 4x - 3.

Vertex Calculation:

• a = -1,
• b = 4,
• c = -3

Vertex x value: x = -\frac{4}{2 \times -1} = 2
Vertex y value: f(2) = -2² + 4 * 2 - 3 = 1
Vertex: (2, 1)

Axis of Symmetry: x = 2

Y-intercept:f(0) = -3
X-intercepts: Solve -x² + 4x - 3 = 0
- (x - 1)(x - 3)= 0

X-intercepts: x = 1 and x = 3

Graph: Plot the vertex (2, 1) the y-intercept (0, -3) and x-intercepts (1, 0) and (3, 0). The Draw the parabola opening downwards.

Example 4: Finding the Parabola for a Given Vertex:(1, -2) and the parabola passes through ( 0, 1).

Vertex Form: f(x) = a(x - 1)² - 2

Substitute the Point (0, 1):

• 1 = a(0 - 1)² - 2
• 1 = a - 2
• a = 3

Function: f(x) = 3(x - 1)² - 2

Graph: Plot the vertex (1, -2) and the given point (0, 1). Draw the parabola opening upwards.

Practical Problems

Questions 1. Graph the quadratic function f(x) = 3x² - 6x + 2 and find its vertex axis of symmetry, and intercepts.

Questions 2. Determine the roots of the quadratic function f(x) = x² + 4x + 4 and sketch the graph.

Questions 3. Find the maximum height reached by the projectile with a height function h(t) = -5t² + 20t + 10.

Questions 4. Solve for x in the quadratic equation 2x² - 5x + 3 = 0 and graph the function.

Questions 5. Identify the vertex of function f(x) = -x² + 6x - 8 and sketch its graph.