Question

match the following; standard form: $y = 2x^2 - 3x + 5$; vertex form: $y = -5(x + 2)^2 - 1$; intercept (factored) form: $y = -2(x + 4)(x - 7)$

Explanation:

Step1: Recall Standard Form

The standard form of a quadratic equation is \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The equation \( y = 2x^2 - 3x + 5 \) is in this form, so it matches Standard Form.

Step2: Recall Vertex Form

The vertex form of a quadratic equation is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. The equation \( y = -5(x + 2)^2 - 1 \) can be rewritten as \( y = -5(x - (-2))^2 + (-1) \), which fits the vertex form, so it matches Vertex Form.

Step3: Recall Intercept (Factored) Form

The intercept (factored) form of a quadratic equation is \( y = a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the roots (x - intercepts). The equation \( y = -2(x + 4)(x - 7) \) can be rewritten as \( y = -2(x - (-4))(x - 7) \), which fits the factored form, so it matches Intercept (Factored) Form.

Answer:

• Standard Form: \( y = 2x^2 - 3x + 5 \)
• Vertex Form: \( y = -5(x + 2)^2 - 1 \)
• Intercept (Factored) Form: \( y = -2(x + 4)(x - 7) \)