Quadratic Graph

A quadratic graph represents the visual shape of a quadratic function, which is a polynomial of degree 2. The general form of a quadratic function is:

f(x) = ax² + bx + c

• Where a, b, and c are constants, and a ≠ 0, x is the independent variable.

The graph of this function is called a parabola, a U-shaped curve that can open upwards or downwards depending on the sign of the coefficient a.

• If a > 0, the parabola opens upwards.
• If a < 0, the parabola opens downwards.

Read More about Graph of Quadratic Equation.

Key Features of Quadratic Graphs

Some of the key features of quadratic graphs are:

• Vertex
• Axis of Symmetry
• Direction of Opening
• X and Y Intercept

Let's discuss these in detail.

Vertex of the Quadratic Graph

The vertex of the quadratic graph is the highest or lowest point on the parabola depending on its orientation. For the function f(x) = ax² + bx + c the vertex can be found using the formula:

x = -b / (2a)

Substitute this x value into the function to the find the corresponding y value. The vertex is given by:

Vertex = -b / (2a) , f(-b / (2a))

Example: For the quadratic function f(x)=x²:

• Find the x-coordinate of the vertex: x = \frac{-b}{2a} = \frac{-0}{2 \cdot 1} = 0
• Find the y-coordinate of the vertex by the substituting x=0 into the function: f(0) = 0² = 0
• So, the vertex is (0,0).

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into the two mirror-image halves. It has the equation:

x = -b / (2a)

Example: For the function f(x) = x² − 4:

• Find the x-coordinate of the axis of the symmetry: x = \frac{-b}{2a} = \frac{0}{2 \cdot 1} = 0
• The axis of the symmetry is x = 0.

Direction of Opening

The direction in which the parabola opens depends on the coefficient a:

• If a > 0 the parabola opens upwards.
• If a < 0 the parabola opens downwards.

Example: For the function f(x) = x²:

Check the coefficient a:

• Here, a = 1 which is positive.

Thus, the parabola opens upwards.

X-Intercepts and Y-Intercepts

• X-Intercepts are the points where the graph intersects the x-axis. Set f(x) = 0 and solve for the x.
• Y-Intercept is the point where the graph intersects the y-axis. Set x = 0 in the function:

Y - Intercept = f(0) = c

Example: For the function f(x) = x² − 1:

Find the x-intercepts:

• Set f(x) = 0: x² − 1 = 0
• Factor the quadratic equation:(x − 1)(x + 1) = 0
• Solve for the x: x = 1 or x = −1
• The x-intercepts are (1, 0) and (−1, 0).

Find the y-intercept:

• Set x = 0: f(0) = 0² − 1 = −1
• The y-intercept is (0,−1).

Plotting a Quadratic Graph

Steps to plot quadratic graph are:

Step 1. Find the Vertex using the vertex formula.

Step 2. Determine the Axis of the Symmetry.

Step 3. Find the X-Intercepts by the solving ax² + bx + c = 0.

Step 4. Find the Y-Intercept by the setting x = 0.

Step 5. Plot the Vertex, Intercepts and Additional Points if needed.

Step 6. Draw the Parabola through these points.

Example: For the quadratic function f(x) = x² − 2x + 1

• Vertex: (1, 0)
• X-Intercepts: (1, 0)
• Y-Intercept: (0, 1)

Using the Vertex Form of a Quadratic Equation

The vertex form of the quadratic equation is:

f(x) = a(x - h)² + k

where (h, k) is the vertex. This form makes it easy to the identify the vertex directly.

Using the Standard Form of a Quadratic Equation

The standard form is:

f(x) = ax² + bx + c

To convert to the vertex form complete the square:

• Factor out a from the first two terms.
• Complete the square inside the parentheses.
• Simplify the equation.

Transformations of Quadratic Graphs

Some of the common transformations of quadratic graphs are:

• Vertical and Horizontal Shifts
• Reflections and Stretching
• Effect of Coefficients on the Graph Shape

Vertical and Horizontal Shifts

• Vertical Shifts: f(x) = ax² + bx + (c + k) shifts the graph vertically by the k units.
• Horizontal Shifts: f(x) = a(x - h)² + k shifts the graph horizontally by the h units.

Reflections and Stretching

• Reflections: If a is negative the graph reflects across the x-axis.
• Stretching: The value of the affects the width of the parabola. The Larger |a| values make the graph narrower while smaller |a| values make it wider.

Effect of Coefficients on the Graph Shape

• Coefficient a: The Determines the direction and width of the parabola.
• Coefficient b: The Affects the position of the vertex along the x-axis.
• Coefficient c: The Affects the position of the parabola along the y-axis.

Applications of Quadratic Graphs

The Quadratic graphs are used in the various real-life scenarios including:

• Projectile Motion: The Modeling the trajectory of the objects in the motion under gravity.
• Optimization Problems: Finding the maximum or minimum values in the business and economics.
• Engineering: Designing the parabolic reflectors and other structures.

Read More: Real-Life Applications of Quadratic Equations

Solved Example: Quadratic Graph

Example 1: Graph: f(x) = x² - 4x + 3

Solution:

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.

Solution:

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.

Solution:

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).

Solution:

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 Questions

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.

Questions 6. Compare the graphs of f(x) = x² + 2x + 1 and g(x) = x² - 2x + 1 and describe their differences.

Questions 7. Find the x-intercepts of the quadratic function f(x) = x² - 3x - 10 and sketch the graph.

Questions 8. Determine the quadratic function that has a vertex at (3, -4) and passes through the point (4, -1).

Questions 9. Calculate the vertex and axis of symmetry for function f(x) = 2x² + 8x + 6 and plot the graph.

Questions 10. Graph the quadratic function f(x) = -2(x - 3)² + 7 and find its vertex and intercepts.

Read More,

• Quadratic Function
• Quadratic Equation
• Standard Form of Quadratic Equation